The American, champion at Roland Garros in 2002 and 2013, spent most of the late evening match on Court Philippe Chatrier trailing, but flicked the switch just in time, to reel off the last six games of the match with a flurry of winners.
Her victory means the 33-year-old is the first woman in the professional era to win 50 matches at each of the four grand slams.
"I don't understand that, because it's just weird to me, because I was thinking 'are you sure Martina (Navratilova) didn't do it or Chrissie Evert?'" said Williams.
Double Australian Open champion Azarenka, coming back to her best after a foot injury dropped her down the rankings in 2014, bossed the match for an hour and after winning the first set looked poised for victory at 4-2 ahead in the second.
But Williams, who stuttered badly against 105th-ranked German Anna-Lena Friedsam in the previous round and had looked subdued against Azarenka, suddenly caught fire.
Celebrating every winning point with clenched fists, 33-year-old Williams broke back and moved 5-4 ahead with an ace.
Azarenka saved three set points in the next game, battling back from 0-40, but Williams landed a fourth opportunity and received a large slice of luck when Azarenka drove a forehand on to the baseline with too much power for her opponent to handle.
Williams argued correctly there had been a late call of 'out' and although the umpire inspected the mark and declared Azarenka's shot had been good, the point was replayed -- much to Azarenka's annoyance.
"He did a late call, he called it out, he didn't put his arm out until like five minutes later. But yeah, I mean, this is tennis," Williams said.
She took full advantage of her good fortune, taking the set with a forehand winner.
Azarenka disappeared for a bathroom break, no doubt to calm down, and moved 2-0 ahead on her return but Williams again raised the intensity level.
Three crunching backhand winners helped Williams get back to 2-2 and, with Azarenka's resolve weakening, she moved in for the kill, swiftly polishing off the last few games.
Her 16th backhand winner completed the comeback and Williams celebrated with a neat pirouette on the baseline.
(Reporting by Julien Pretot; Editing by Martyn Herman and Ian Chadband)
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