The seventh seed from Britain won 3-6 6-3 6-3 4-6 12-10 after 40 minutes' play on Sunday.
The thrilling tennis of late Saturday continued with no sign of the leg problems which had Murray grimacing in the Paris gloom.
The pair resumed at two sets all and 7-7 and Murray had a match point at 9-8. Kohlschreiber saved it with a brave overhead after a great rally but Murray made no mistake on his second, scything a backhand return past his opponent.
Kohlschreiber had shown tremendous composure as he served to stay in the match six times but finally succumbed at the seventh time of asking, although he did have a break point of his own during Sunday's tense finale.
Murray plays 24th seed Fernando Verdasco of Spain who beat 12th seed Richard Gasquet of France in another resumed match on Sunday.
"Last night was tough as I did a lot of running," said Murray. "Yesterday I was cramping so maybe it was a good thing the match was stopped.
"I didn't sleep much last night and woke up a few times, I was very nervous.
"I just tried to do my usual preparations but it is not easy coming back 7-7. Considering the circumstances today I thought we played some very good tennis."
Kohlschreiber was sanguine about his defeat. "I think I did a very good job, very happy, pleased with my game. I tried a few things. They worked out pretty well. At the end, there were just a few points that make the difference.
"Unfortunately someone has to lose. But it was just yesterday and today was a great experience for me. He's a huge fighter. He never gives up."
Murray missed last year's French Open with a back injury.
(Reporting by Robert Woodward; editing by Martyn Herman)
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