The match between Wimbledon champion Murray and the 28th-seeded Kohlschreiber was suspended because of fading light Saturday.
Kohlschreiber missed a chance to break Murray in the 19th game of the decider.
Murray, a two-time Grand Slam champion, crouched down and clenched both fists after hitting a two-handed backhand winner on Kohlschreiber's second serve with the score at 15-40 in the 22nd game.
"Last night was extremely tough for me. I did lot of running," Murray said. "Considering the circumstances, today I thought we both played some good points. We both came out probably pretty nervous."
Murray, the 2012 U.S. Open champion, made light of a disturbed night's sleep of barely five hours.
"I woke up quite a few times," he said. "I was kind of ready to play at 4 or 5 in the morning."
He will get a longer rest before playing No. 24 Fernando Verdasco of Spain, who easily beat 12th-seeded Frenchman Richard Gasquet 6-3, 6-2, 6-3. Murray leads Verdasco 9-1 in their career head-to-heads but they have never met on clay.
The 18th-seeded Canadian wasted no time, opening up a 5-0 lead in 16 minutes. Bouchard, who reached the Australian Open semifinals this year, next plays either No. 14-seeded Carla Suarez Navarro of Spain or the unseeded Ajla Tomljanovic. They were playing later Sunday.
"I really believe in my skills. I believe I can play with the best girls out there," Bouchard said. "I was really mentally prepared for anything, for a battle. I think that mindset kind of helped me."
In the men's fourth round, Tomas Berdych defeated 10th-seeded John Isner 6-4, 6-4, 6-4. The sixth-seeded Czech next faces either 17-time Grand Slam champion Roger Federer, seeded fourth, or No. 18 Ernests Gulbis of Latvia. They were playing their match later Sunday.
With the top three women's seeded players out, Maria Sharapova remains the favorite to win the tournament for the second time. The seventh-seeded Russian later played Samantha Stosur.
"I think it just makes the draw really interesting when some of the top names are out and we see some new names that we haven't seen that much," Bouchard said.
Trailing 5-2, Kerber played with the strings of her racket as she hunched forward on her chair, her head bowed. Bouchard, by contrast, sat upright, taking a few deep breaths to compose herself before serving out the match. She clinched victory on her first match point when Kerber — a two-time Grand Slam semifinalist — made another unforced error, this time as her wild forehand sailed out.
"I was not in the game. I was trying, but I had no rhythm," Kerber said. "I make so many mistakes."
The same could be said of Isner, who snapped his racket in frustration toward the end of the match.
The 28-year-old Berdych improved to 5-2 against Isner in their career head-to-heads, and it was his third straight win against him. Berdych has won all three of their meetings on clay, and now beaten Isner twice at the French Open. He also won in straight sets when they met in the third round in 2010, the year Berdych reached a career-best semifinal at Roland Garros and made the Wimbledon final.
The 29-year-old Isner was the last American man in the field and was trying to be the first U.S. man to reach the French Open quarters since eight-time Grand Slam champion Andre Agassi in 2003.
"I was a little slow out there," Isner said. "He was a lot better than me today."
In men's fourth-round action later Sunday, second-seeded Novak Djokovic was playing 13th-seeded Jo-Wilfried Tsonga.