Visiting Bournemouth had shocked their Etihad Stadium hosts with a Callum Wilson equaliser a minute before the break to cancel out Bernardo Silva's 16th-minute opener.
Bournemouth had strong claims for a penalty turned down after Fernandinho appeared to push Tyrone Mings in the area, but Gundogan doubled City's lead in the 79th minute, coolly finishing from David Silva's low cross.
Silva scored the opening goal as Pep Guardiola's side beat Bournemouth 3-1 on Saturday, but it was not a comfortable win as the Cherries pushed Manchester City late in the first half and towards the end of the game.
City remain top of the league heading into the midweek fixtures, but Liverpool's incredible late victor against Everton in Sunday's Merseyside derby means the gap is still just two points. I play nine games but many other contenders, our opponents, play nine games too.
"The doctors told me Sergio can not play", Guardiola told a news conference.
The England winger's ninth goal of the season owed more to opportunism than ingenuity, however, as Begovic clumsily spilt Danilo's shot for him to find the roof of the net.
It seemingly needs a sleepy start to a game to jolt Manchester United's players into action these days.
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Today, we pick out three players in the Premier League now who can achieve a lot more if they move on to another club.
The goal was the Belgian striker's first for his club since the middle of September, a drought lasting 981 minutes.
He is the Premier League's joint-leading English scorer, his eight goals matching Harry Kane's tally after the weekend games, and is joined by two others on top of the assists table.
Ander Herrera equalised before halftime to bring United fully back into the contest but the match petered out into a draw, a result that will not please either manager much.
Palace leapt above Huddersfield in the standings by beating Burnley at Selhurst Park to lift the pressure on manager Roy Hodgson.
The Hornets head into their midweek match with only two triumphs to show from their last 10 Premier League matches (W2-D2-L6), whilst suffering defeat in four of their five contests against top-half teams.
They are just one point behind eighth-placed Leicester City, which defeated Watford 2-0 thanks to first-half goals from striker Jamie Vardy and midfielder James Maddison.
Watford's misery was complete when Etienne Capoue was sent off in the 90th minute for a lunge on Kelechi Iheanacho.