Murray earned the only two breaks of the third set and closed out the win when Kohlschreiber's forehand went wide, one of 35 unforced errors by the German. The temperature topped 90 degrees, unusually warm for this time of year.
Cool mornings, hot afternoons and warm evenings have made for changing court conditions and ball speed in the desert, and Murray was affected by them.
"When you are playing in those matches during the heat of the day, you need to trust your shots. You need to go after them, because if you back off and try and sort of guide the ball in these conditions, it doesn't work," he said. "During the day it's ridiculous how high the balls bounce and how quick they move through the air and jump off the court."
Murray moved on to a fourth-round match against Adrian Mannarino, who beat 14th-seeded Ernests Gulbis, 6-4, 6-4.
Fifth-seeded Kei Nishikori was stretched to three sets before overcoming Fernando Verdasco, 6-7 (8), 6-1, 6-4. Nishikori double-faulted twice in the final game before advancing to the fourth round for the first time in his seventh appearance at Indian Wells.
"Third set, it could go both ways, but I got first break," Nishikori said. "I really served well. Until last game I didn't face break points. It was still close the last game, so really happy to beat Fernando."
John Isner beat 18th-seeded Kevin Anderson, 7-6 (8), 6-2, setting up a possible fourth-round match against top-ranked Novak Djokovic, who played Albert Ramos-Vinolas in a night match.
On the women's side, former Indian Wells winner and No. 4 seed Caroline Wozniacki was beaten 6-4, 6-4 by 31st-seeded Belinda Bencic a week after the Dane won her 23rd career singles title at Kuala Lumpur.
Bencic earned her first win over a top-five player and at 18 became the youngest player to reach the fourth round this year. Last year in Istanbul, Bencic failed to win a game off Wozniacki.
"In Istanbul I had maybe too much respect and I was afraid, nervous," she said. "Today I really had a good game plan. I served well kind of, and had sometimes some easy points on my serve because of that."
Jelena Jankovic, who won here in 2010 and spent 18 weeks at No. 1 in the world, outlasted Madison Keys, 5-7, 6-4, 6-3 despite the Serb having just 13 winners and 42 unforced errors.
Keys, a 20-year-old American coached by former top-ranked Lindsay Davenport, had her chances to take control, leading 3-1 in the final set. She hit 39 winners, but came undone with a whopping 64 unforced errors.
Qualifier Lesia Tsurenko defeated 20th-seeded Alize Cornet, 7-5, 1-6, 6-2.
Eugenie Bouchard defeated CoCo Vandeweghe 6-3, 6-2 to reach the fourth round.
The sixth-seeded Canadian faced the only break point on her serve of the match in the last game, but Vandeweghe netted a forehand. Bouchard hit a forehand winner on her second match point. Vandeweghe had 24 unforced errors in the third-round match, the farthest the American has gotten at Indian Wells in four tries.
"I felt very solid," Bouchard said. "That's important against a player who can have big weapons. She had some great serves and some great forehands. I was going to try to neutralize that and take my chances when I had them."
Bouchard had plenty of support in the crowd from the many Canadians who spend the winter in the desert. They waved red and white Maple Leaf flags in what seemed like a home match for Bouchard.
Maria Sharapova and Victoria Azarenka meet in an evening match between former No. 1 players.