They took the lead less than 10 minutes later, however, when Ribery volleyed into the back of the net from close range after receiving a good pass from Arjen Robben.
Phillip Lahm then got in on the act on 23 minutes when the German defender capitalised on a loose ball inside the box before slotting into the back of the net with ease.
Monchengladbach did have chances of their own in the first period, but Bayern had their third less than three minutes later when Thiago Alcantara chested a Jerome Boateng cross into the bottom corner.
Ribery had the first real chance of the second period, but the Frenchman could only find the arms of Christofer Heimeroth in the Monchengladbach goal.
Tony Jantschke threatened to reduce the deficit to just one goal after 38 minutes, but his header moved high and wide of the crossbar, before Ribery was again denied by Heimeroth down the other end moments later.
Bayern had their fourth after 41 minutes, though, when Robben met a cross from Ribery before thumping a header into the back of the net.
Robben and Ribery both came close to making it five as both sides make changes in the latter stages, but it was Thomas Muller that grabbed Bayern's fifth with the final kick of the match.