Former Hamburg striker Sidney Sam struck on the half-hour for Leverkusen but Arturo Vidal's own goal briefly gave the hosts some hope before the Chilean made amends in the 61st minute.
Renato Augusto added two more to pile on the misery for Hamburg, who cut the deficit in the 79th minute through Eljero Elia.
The win lifted Leverkusen to 32 points from 16 games, eight behind leaders Borussia Dortmund, who take on Werder Bremen later on Saturday.
Bayern took the lead through a thundering long-range drive from Hamit Altintop against the run of play but wasted several chances early in the second half.
They needed a penalty and a red card, awarded for keeper Thomas Kessler's foul on Thomas Mueller, to make things safe.
Philipp Lahm converted from the spot and with 10 men St Pauli were finished. Franck Ribery completed a victory that lifted them to 26 points.
Surprise package Hanover climbed to 31 points after sinking VfB Stuttgart 2-1 on Friday for their 10th win of the season, one more than in the whole of last season in the league.
Mainz 05, on 30 points, host Schalke 04 on Sunday
