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 33rd career title, a welcome boost ahead of the French Open -- the second Grand Slam of the season that is played on clay.
Murray called the title "incredible."
Murray fired 17 aces to Kohlschreiber's 11. Otherwise, little separated the two players.