Frank Lampard's side have spent lavishly in this transfer window with a view to closing the 33-point gap which opened up between these sides last season but couldn't lay a glove on the title holders in the first half. Their task was then made so much harder in stoppage time when Christensen was sent off for hauling Mane to the ground.
That paved the way for Liverpool - who brought on new signing Thiago at half-time - to take complete control after the break and they did just that with Mane heading in Roberto Firmino's cross (50) and then chasing down Kepa's casual pass out to slot in a second (54).
Timo Werner's perseverance gave Chelsea hope of an unlikely comeback when he was tumbled in the penalty area by Thiago, but Jorginho's spot-kick was saved and Jurgen Klopp's side marched to a second straight win with a daunting performance for their title rivals.
Instead, with Chelsea - missing new arrivals Ben Chilwell, Hakim Ziyech, Thiago Silva and key man Christian Pulisic through injury - happy to sit back and wait for opportunities to counter through Werner, there was just one shot on target during the opening period.
That came 13 minutes in, with a reminder of the air of uncertainty around Kepa. The Chelsea goalkeeper misguidedly chased a deflection to the side of his area, was beaten to the ball by Mohamed Salah and looked on thankfully as Christensen got back to block Roberto Firmino's shot.
But Christensen went from hero to villain for Chelsea and left them facing the entire second half with 10 men after rugby-tackling Mane to the floor as he ran through on goal. Referee Paul Tierney initially brandished a yellow card but upgraded it to a red after consulting the pitch-side monitor.
Chelsea: Kepa (4), James (5), Christensen (3), Zouma (5), Alonso (5), Jorginho (5), Kante (6), Kovacic (6), Mount (6), Havertz (5), Werner (6).
Subs: Tomori (6), Abraham (n/a), Barkley (n/a)
Liverpool: Alisson (8), Alexander-Arnold (6), Fabinho (8), Van Dijk (7), Robertson (7), Wijnaldum (8), Keita (5), Henderson (6), Salah (6), Mane (9), Firmino (7).
Subs: Thiago (6), Milner (6), Minamino (n/a)
Man of the match: Sadio Mane (Liverpool)
While Chelsea subbed off expensive new signing Kai Havertz for defender Fikayo Tomori, Liverpool's new man Thiago came on for his debut at the start of the second half and typically took charge of the midfield as the visitors asserted their control.
The Reds made that superiority count just five minutes after the restart when Firmino played a one-two with Salah around Marcos Alonso and Mane headed home the Brazilian's cross.
Just four minutes later their advantage was doubled when Mane chased down the ball after giving up possession and intercepted Kepa's pass before slotting past the stranded and under-pressure goalkeeper.
Despite Liverpool's authority, Chelsea came close to getting back into the match with a couple of rare forays forward. First Mason Mount saw a dipping effort clip the roof of the net and then Werner won a penalty from Thiago's clumsy clip but Jorginho failed to find the net from the penalty spot for the first time in the Premier League.
Salah almost made it three with the final kick of the game but shot straight at Kepa, but regardless of the modest scoreline, this was a resounding response from the champions to their ambitious opponents.
Man of the Match: Sadio Mane
Sky Sports' Jamie Carragher:
"I said he was the best player last year. To be fair, Kevin de Bruyne got player of the year and I think Jordan Henderson got the other one, but certainly for me I thought Sadio Mane was Liverpool's best player and he'd have been very close in the running.
"He's certainly one of the best left-sided attacking players in world football. I love him. I say it all the time and it's well-documented. I always go back to Mane as he was the first big signing for Jurgen Klopp.
"That signing got Liverpool into the Champions League in Liverpool's first full season, and without that, I don't think it goes on from there. Then, other players come, more revenue comes into the club but he was the one who came in and has been a constant since day one with Jurgen Klopp and is an absolute superstar."
What the managers said...
Chelsea boss Frank Lampard: "[I was happy about] our defensive game. I wanted them to be better on the ball. We showed moments on the counter-attack where we dangerous. But to limit them to the opportunities they got in the first-half, there were lots of things I was pleased about. The red card changed the team talk. They can be given or not - I'm not going to complain too much.
"I'm disappointed to lose three points but I'm happier today in many ways than I was in Brighton. Today I saw spirit, a desire to defend by a lot of our players. The players should take heart.
"It's how we go forward from here now. Every team that has been successful in the modern day has had time, has had a process. Our process starts here. Now it's a new start, our objectives are higher, our expectations are higher, fine. But it will take some time."
Liverpool boss Jurgen Klopp: "Super game. Really good game from both teams. I like our game a lot. Chelsea are a top side. Our own football I liked in the first half, really good situations, passed through the line but we didn't score.
"After the red card, the second half is a different game. We kept the ball rolling, let them run, scored the goals, conceded a penalty, Ali saved it, so all good. Everybody had a hand in the result."
Analysis: Kepa 'going through a nightmare'
Sky Sports' Jamie Carragher:
"As players, you go through tough times at different times in his career and you actually feel a bit sorry for Kepa now. There's a new goalkeeper that looks like he's coming in so I don't think there's much more we can say about the goalkeeper.
"It must be tough for him and he's going through a really tough period. Our job is to be critical at times, but I remember when I went through a similar period earlier in my career. It wasn't going great and he's going through his own nightmare at the moment."
Analysis: No doubt it was a red card
Sky Sports' Jamie Redknapp:
"I have no doubt in my mind that Mane would've got to the ball and taken it away from Kepa so it's definitely a red card. Christensen is static and by the time he moves it's too late. It gave Liverpool such momentum for the second half."
Analysis: Performance of champions from Liverpool
Sky Sports' Jamie Carragher.
"Coming into this game there were a few question marks against Liverpool. There was the feeling this would be a tough game.
"But they put in the sort of performance that champions do."
- Liverpool were the first reigning champions to win away at Chelsea in the Premier League since Manchester United in April 2002, with the Blues unbeaten in 13 such matches since then before today (W7 D6).
- Chelsea have both had a player sent off and missed a penalty in a Premier League for the first time since February 1999 against Blackburn, when Gianluca Vialli was sent off and Frank Leboeuf missed from the spot.
- Excluding shootouts, Jorginho missed a penalty for the first time in nine attempts for Chelsea in all competitions, while this was Alisson's first penalty save for Liverpool since joining the club (three faced). Indeed, the penalty was Alisson's first save of any kind in this season's Premier League.
- Despite only coming on as a second half substitute, no Chelsea player completed more passes (75) or had more touches (89) than Liverpool debutant Thiago. His 75 successful passes was the most by a player to play 45 or fewer minutes in a Premier League game since 2003-04 (when full passing data is available).
Chelsea face Barnsley in the Carabao Cup on Wednesday evening and then travel to West Brom on Saturday, live on Sky Sports. Liverpool face Lincoln in the Carabao Cup on Thursday, live on Sky Sports, and then take on Arsenal on Monday Night Football in the Premier League on September 28, also live on Sky Sports.