The Norway striker, who has now scored 42 goals in 37 games since joining Pep Guardiola’s side last summer, struck twice in 179 first-half seconds to put the hosts in control.
Haaland poked the ball past goalkeeper Bailey Peacock-Farrell for his first before curling a superb second into the far side of the goal.
Former City captain Kompany’s runaway Championship leaders Burnley had given a good account of themselves, but Haaland grabbed his third before the hour mark after Phil Foden’s shot came back off the post.
Julian Alvarez then slotted in Kevin de Bruyne’s pass, and after substitute, Cole Palmer added a fifth the Argentine fired in his second from another De Bruyne assist.
The draw for the FA Cup semi-finals takes place on Sunday following Brighton’s tie against Grimsby Town (14:15 GMT), live on BBC One.
