Murray, seeded fourth, raced through the first set in 25 minutes, but Kohlschreiber hit back in the second and it took almost two hours for the number four seed to wrap up a 6-1 3-6 6-1 win.
Murray admitted he was not at his best in the hot conditions but felt like he was in control of the match.
“In the first set I was dictating the rallies, the ball was bouncing high and I was pushing him away,” Murray said.
“In the second set there was a break point at 3-3, it gave him confidence, but in the third set I started dictating points again and getting good height on the ball.”
Murray conceded only one of the seven break points he faced but committed five double faults to help Kohlschreiber stay in the match.
“I feel I can serve better, serve smarter than I did today,” he added.
“Philipp’s not one of the taller players so when I can get it to bounce high it makes it tough for him on the return.
“When I was getting myself in position to dictate I wasn’t letting up, I kept him moving around.
???I maybe didn’t hit a lot of winners but there were a lot of forced errors and in conditions like this that’s important.”
Murray will next face Frenchman Adrian Mannarino, who reached a career high No. 36 in the world rankings earlier this season and who beat Ernests Gulbis 6-4 6-4 in the third round.
“He’s improved a lot over the last 18 months and he’s at about his highest ranking right now,” Murray said of his Mannarino.??
“He’s had a couple of good wins here, he’s a talented lefty and like most French players he plays with a bit of flair on court.
“I’m sure there’ll be some interesting points and I’ll have to play well to beat him. He’ll have confidence right now.”