Haaland had been sidelined for nearly two months by a foot injury, but City barely missed the presence of the Norwegian as they won nine and drew one of the 10 games when he was absent.
Julian Alvarez again stepped up in Haaland's absence as the Argentine struck twice on his 24th birthday to effectively end the game at the Etihad as a contest inside 22 minutes. The Argentine headed in from Matheus Nunes' cross and then finished off a clever free-kick from Kevin De Bruyne.
Rodri made it 3-0 inside the first minute of the second half, but Pep Guardiola made the expectant home crowd wait until 15 minutes from time for Haaland's introduction.
Rather that a goalscoring return from last season's Golden Boot winner, it was the Clarets who struck next to end a run of 22 consecutive City goals in league meetings between the sides.
Ameen Al Dakhil struck four minutes into stoppage time, but it was no more than consolation for the Clarets, who remain second bottom and seven points adrift of safety.