City's Brazilian striker had already seen an effort ruled out for offside when he put them ahead with a curling effort from the left-hand side of the penalty area on 24 minutes.
He doubled the lead in the 50th minute with his fifth Premier League goal of the season, flicking in Bernardo Silva’s right-wing cross from close range.
Rodri made the match safe midway through the second half with a crisp volley from the edge of the area, with Riyad Mahrez adding the fourth, his 50th Premier League goal, from 20 yards out.
Burnley substitute Robbie Brady scored a late consolation for the hosts.
The win takes City back up to second, eight points behind Liverpool, while Burnley slip to 11th on 18 points.
Burnley: Dec 7 v Spurs (A)
Man City: Dec 7 v Man Utd (H)