The top-seeded Murray was the steadier player in the decisive tiebreaker and clinched the close match when Kohlschreiber sent a backhand long on Murray's second match point.
The final was rained out on Sunday, with former two-time Munich champion Kohlschreiber leading 3-2 on serve, and postponed until Monday.
Murray was in his first final on clay and won his 32nd career title, a welcome boost ahead of the French Open - the second Grand Slam of the season that is played on clay.
The Scot, ranked No, 3 in the world, became the first British man to win a title on clay since Buster Mottram in 1976.
''It's incredible, Murray said. ''Philipp was very tough to beat.''
Murray fired 17 aces, while Kohlschreiber had 11. Otherwise, little separated the two players. Kohlschreiber was seeking his third title in Munich after winning in 2007 and 2012.
After trading early breaks in the second set, Murray had a chance to go 5-3 up but the German saved three break points to level at 4-4.
Kohlschreiber did get the break for a 6-5 lead and took the set with a backhand winner.
Both players held serve in the final set, with Murray earning three match points in the tiebreaker at the end of a long rally that finished with an error by Kohlschreiber.
''I was lucky on a couple of passing shots late in the match,'' Murray said.