Pep Guardiola’s side had been overtaken as leaders by Liverpool’s 2-0 win over Watford at Anfield earlier in the day, but the champions responded and maintained their record of winning all 23 matches they have led this season.
The opener came in just the fifth minute as Raheem Sterling cushioned the ball into the path of De Bruyne, who lashed in his 10th Premier League goal of the season, reaching double figures in the competition for the second time.
Twenty minutes later Sterling provided another assist, playing a one-two with De Bruyne and cutting the ball back for Gundogan to finish.
City missed chances to cut into the five-goal difference by which they trail Liverpool, coming closest when Gabriel Jesus’s shot was deflected on to a post.
Guardiola’s side are on 73 points, one ahead of Liverpool ahead of next weekend’s top-of-the-table meeting at Etihad Stadium.
Burnley remain 19th on 21 points, four from safety and having played an extra match than 17th-placed Everton.