City goalkeeper Ederson was stretchered off in a neck brace after being caught in the face by Mane's boot as he raced out to head the ball away, with the Liverpool winger dismissed for the clash late in the first half at Eastlands.
City already led 1-0 through Sergio Aguero's opener at that point and braces from Gabriel Jesus and substitute Leroy Sane completed City's biggest win over Liverpool since 1937.
City are unbeaten in 12 Premier League games, their best run under Guardiola, while Argentina striker Aguero is now the outright leading non-European goalscorer in Premier League history with 124 goals.
"After 10 v 11 it was a little bit easier for us and we played the second half the way we wanted to," said City manager Pep Guardiola.
Liverpool boss Jurgen Klopp said: "The decisive decision in the whole game was the red card. I don't think it was a red card. Sadio didn't see the goalkeeper."
After a blistering start to the season, United conceded for the first time in the league this term when Eric Maxim Choupo-Moting swept home from Mame Biram Diouf's 43rd minute cross to notch his first Stoke goal.
Jose Mourinho's side drew level at the bet365 Stadium as Henrikh Mkhitaryan's corner in first half stoppage-time was headed goalwards by Paul Pogba before deflecting into the net off Marcus Rashford.
The visitors moved ahead in the 57th minute when Mkhitaryan's defence-splitting pass was converted by Romelu Lukaku after the Belgian forward's initial effort was saved by Jack Butland.
But when United defender Phil Jones stumbled at a Stoke corner, Cameroon forward Choupo-Moting was unmarked to head the equaliser in the 63rd minute.
Despite dropping points for the first time in four matches, United remain top of the table, above neighbours City on goal difference.
"It was a difficult match. My players, after the international break, are not the same," Mourinho said.
"I'm not satisfied with a point because we were the team closer to winning."
At the Emirates Stadium, Arsenal eased the pressure on Arsene Wenger with a vital 3-0 win against Bournemouth.
Subjected to stinging criticism following Arsenal's damaging defeats against Stoke and Liverpool, it took just six minutes for Danny Welbeck to lift Wenger's gloom as the Arsenal striker headed home from Sead Kolasinac's cross.
France forward Alexandre Lacazette doubled Arsenal's lead in the 27th minute, bagging his second goal since his pre-season move from Lyon with a curling strike from the edge of the area.
Welbeck's second goal, from Aaron Ramsey's 50th minute pass, secured Arsenal's first win since the opening weekend.
Alexis Sanchez came off the bench in the closing stages for his first Arsenal appearance this season following the collapse of his mooted move to Manchester City on transfer deadline day.
"It was a serious performance. We played well all over the pitch," Wenger said.
Harry Kane reached his Tottenham century as the England striker's brace inspired an impressive 3-0 win at Everton.
Kane bagged his 100th Tottenham goal – and first this season – in the 28th minute, looping a long-range cross-shot over Everton goalkeeper Jordan Pickford.
Christian Eriksen increased Tottenham's advantage with a close-range strike in the 42nd minute after Pickford spilled a Ben Davies shot.
Kane bagged his second from Davies's 46th minute cross to kill off Everton.
Chelsea survived a tense finish as the champions made it three successive wins with a 2-1 success at Leicester.
Antonio Conte's side took the lead through Alvaro Morata's third goal for Chelsea as the Spain striker peeled away from Wes Morgan to meet Cesar Azpilicueta's cross with a bullet header in the 41st minute.
N'Golo Kante was a key member of Leicester's title-winning team two years ago and the Chelsea midfielder returned to haunt his old club with a 50th minute long-range strike.
Jamie Vardy got one back for Leicester with a penalty and Conte said: "We must be delighted. We were in total control and could have scored more."
Brighton secured their first top-flight win for 34 years, with Pascal Gross's double clinching a 3-1 win over West Bromwich Albion at the Amex Stadium.
Watford handed Southampton manager Mauricio Pellegrino his first league defeat, 2-0 at St Mary's, thanks to goals from Abdoulaye Doucoure and Daryl Janmaat.