Murray was rocking, but saved three break points at the start of the fourth set, one with a sublime half-volley, and gradually seized control again to prevail 6-2 2-6 4-6 6-2 6-3 in three hours 34 minutes.
"I lost my way on the court today, you know, for quite a while," Murray, who joined the crowd in applauding Borgue, France's 16th ranked player, told reporters.
"I was struggling to win points at one stage. I couldn't see where his shots were going."
"You can't play too many matches like this if you want to go far in this tournament," added Murray, who has already spent seven hours and 15 minutes on the clay.
Murray's opponent in the third round, Ivo Karlovic, was involved in the day's other compelling drama.
The big-serving Croatian, another 37-year-old defying age, beat Australian wildcard Jordan Thompson 6-7 (2) 6-3 7-6(3) 6-7(4) 12-10 to become the oldest man to reach the third round of a slam since Jimmy Connors in 1991.
"It's the only time when being old is okay," 27th seed Karlovic, who has boomed down 72 aces in two rounds, said of his milestone victory.
Like Murray, third seed Wawrinka suffered a first-round fright in a five-setter against Lukas Rosol.
He was still not at his best on Thursday against Japan's Taro Daniel, but having saved two set points in the opening tiebreak he won 7-6(7) 6-3 6-4.
"If you look at the first two matches, for sure I'm not playing my best tennis, but I know I have my best tennis in me," the 31-year-old said. He plays Frenchman Jeremy Chardy next.
Nishikori's progress has been impressive so far and he was too good for Russian Andrey Kuznetsov, winning 6-3 6-3 6-3.
After a flurry of falling women's seeds in round one, order was restored as the sun finally appeared in Paris.
Radwanska bested Caroline Garcia 6-2 6-4 and Muguruza, trying to become the first Spanish woman to win the title since Arantxa Sanchez-Vicario in 1998, thrashed Myrtille Georges 6-2 6-0.
Romania's Halep, runner-up to Maria Sharapova in 2014, took a while to tame Kazakhstan's Zarina Diyas to win 7-6(5) 6-2 and Czech Lucie Safarova, beaten by Serena Williams in the final last year, beat Swiss Viktorija Golubic 6-2 6-2.
Australia's Samantha Stosur did things with relative ease - breaking her Chinese doubles partner Shuai Zhang five times in a 78-minute 6-3 6-4 victory.
Stosur will next face 11th seed, and last year's runner up, Safarova, on Saturday.
The standout clash of the third round will be Australia's Nick Kyrgios against home favourite Richard Gasquet, after both won convincingly on Thursday.