Matt Scott's try and accurate kicking from Jaco van der Walt had visitor Edinburgh 10-0 up by the 20th minute, yet Munster fought back to level the scores heading into the break.
Full debutant Ben Healey kicked five points in Munster's first-half comeback, and two more penalties after the break had the home side 16-10 up.
However, Simon Hickey's penalty moved Edinburgh to within touching distance, with Eroni Sau's try ultimately proving decisive for Richard Cockerill's side as the visitors claimed their first win in Cork since 2004.
Elsewhere, Ulster returned to winning ways in style as it beat Scarlets 29-5 thanks to an exceptional first-half display.
Having been defeated by Munster last time out, Ulster made light work of Scarlets — second in Group B — by running in three tries inside the opening 16 minutes.
Robert Baloucoune added a fourth try by the 27th minute, after Uzair Cassiem had been sent to the sin bin, and though Scarlets did grab a consolation try, it was Ulster that had the final say when Matt Faddes went over.