The Wimbledon champion will face another clay court specialist in the form of Fernando Verdasco on Monday for a place in the quarter-finals after he completed a 6-3 6-2 6-3 win over Richard Gasquet.
"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.
"I need to recover as best as I can for the next match. Hopefully, I will pull up as best as I can so I'm ready to fight for a place in the quarters."
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.
The thrilling tennis of late Saturday continued with no sign of the leg problems which had Murray grimacing in the Paris gloom.
Resuming 7-7 after the match was halted on Saturday night due to bad light, Germany's Kohlschreiber - the 28th seed - was forced to save a match point on his first service game with a fine overhead winner preventing Murray from sealing a 10-8 win.
Murray was forced to save a break point of his own before some superb ground strokes brought up another two match points on his opponent's serve to lead 40-15 in the 22nd game.
He finally secured his progress with a thumping backhand winner flying beyond Kohlschreiber before he had time to react.
SATURDAY'S ACTION BEFORE BAD LIGHT STOPPED PLAY
Kohlschreiber won his last service game, as the sun dipped below the Paris horizon, to leave the fifth set at 7-7. The match will be finished on Sunday on the same Suzanne Lenglen court.
After losing the first set, the British seventh seed appeared to be cruising to victory in the fourth set when he led 4-2. But suddenly Murray's touch deserted him and the German took full advantage to take the match into a pulsating fifth set.
After losing the first set, Murray took a set-winning 3-0 lead in the second. He broke the German in the fifth game of the third set, then wasted three set points before taking the lead in the match with a superb backhand down the line.
Murray appeared in total control after six games of the fourth but gave back his break of serve, and Kohlschreiber, seeded 28th, took advantage of more slack play from his opponent to break for 5-4 and win the set.
Again in the final set, Murray, who had courtside treatment on an unspecified thigh problem as the match approached its climax, broke early. But he allowed his opponent to break back, and then take a 3-2 lead with another break.
Murray came back to 3-3 with a fine backhand cross-court winner, setting the scene for a series of heart-stopping games in fading light as Kohlschreiber tried to take advantage of any problems with Murray's mobility by moving him around the court.
Murray's ability to mix the sublime with the ordinary was demonstrated when he served at 6-6. A double fault was followed by an ace before a driving backhand pass by his German opponent set up a break point.
But Murray replied with his own scintillating backhand winner before clinching the game with a fine drop shot. Kohlschreiber then held his serve after Murray netted a forehand to send the match into a second day.
Watch all the action from the French Open on up to court seven on the
Results from the French Open Men's Singles matches on Sunday
2-Novak Djokovic (Serbia) beat 13-Jo-Wilfried Tsonga (France) 6-1 6-4
18-Ernests Gulbis (Latvia) beat 4-Roger Federer (Switzerland) 6-7(5) 7-6(3) 6-2 4-6 6-3
6-Tomas Berdych (Czech Republic) beat 10-John Isner (U.S.) 6-4 6-4 6-4
8-Milos Raonic (Canada) beat Marcel Granollers (Spain) 6-3 6-3 6-3
7-Andy Murray (Britain) beat 28-Philipp Kohlschreiber (Germany) 3-6 6-3 6-3 4-6 12-10
24-Fernando Verdasco (Spain) beat 12-Richard Gasquet (France) 6-3 6-2 6-3
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