Murray holds a 6-3 edge over the giant Argentine, notably downing Del Potro in a gruelling Olympic final previous year to claim a second straight singles gold medal.
Del Potro's will was broken after that as he lost all six games in the third set to hand Murray a 7-6 (8), 7-5, 6-0 victory.
Del Potro is still working his way back to his rightful ranking after the wrist problems that nearly ended his career, and the big question was how he would pull up physically.
When asked why he kept putting his finger to his lips, Murray said: "I don't know why I was doing it".
"To be playing him this early on in the slam is obviously not easy, but it can be a very positive thing".
You play someone that good, maybe you're a little bit more switched on.
Wawrinka, owner of three major championships in all, will play for a quarterfinal berth against No. 15 Gael Monfils or No. 24 Richard Gasquet, a pair of Frenchmen whose third-round match was interrupted by rain in the first set Saturday. "You never know when you can win a set or a match, and that happened today".
"The tactical side is very important for me". It's irrelevant. 'I was thinking what was going on out there and I was thinking between the points.
"I'm starting to feel better".
"That's been the case so far". "I started playing much quicker".
In that respect, what the people in your box do could have an impact on your concentration, but from a player's perspective that time between points is all about preparing for the next one.
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World number two tennis player Serena Williams is the only woman on the list, in 51st, with $27m (£20.92m) made a year ago . Woods ranks No 17 this year, while Mayweather did not make the cut, having had no fights over the past 12 months.
The tie-break was a mini-match in itself, with Murray opening up a 6-4 lead, missing both set points then facing another after Del Potro pulled off a miracle forehand victor off a Murray pass. Whoever had won that first set had big momentum.
He had two set points at 5-3 before Murray clawed it back level by converting his sixth break point.
"It was a really great first set and I'm really pleased with how I battled", said Wawrinka, who was aided by a Fognini knee injury.
While Khachanov is through to the last 16 at a slam for the first time, it is seven years and 25 tournaments since Murray failed to at least make the fourth round.
They each had a significant win past year with Murray taking gold at the Rio Olympics final and Del Potro edging a five-set thriller when Argentina beat Britain in the Davis Cup semifinals.
Andy Murray benefited from a double-fault by Juan Martin del Potro in the opening tiebreaker and then looked more and more like the No. 1 player that he is during a 7-6 (8), 7-5, 6-0 victory at the French Open on Saturday.
"It's slow and heavy and coming back in these conditions is hard".
The 30-year-old arrived at Roland Garros in disarray, with just one semi-final appearance to show from four clay-court tournaments.
Playing Paris for the first time in five years, del Potro had his moments-and chances to snatch a one-set lead.
He said: "I think that is what we are looking for, to play on the big arenas like here, centre court, and to play against the world No 1".
Del Potro, whose groin caused him trouble in the last match, struggled when stretched wide or in hunting down short balls. Juan Martin kicked off the match in a better fashion, breaking in game 3 for a 4-2 lead, and he had a set point in game 9 on return. He was a real number one. "He has a pretty good demeanour on the court as well, for being very young". "He has all the shots, but also is great mentally". I think my coach, he was watching his match, also.
He said: "I already congratulated him with past results".
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