The Swiss third seed won 6-2, 6-3, 6-3 against the unseeded Mayer to become the first man to book his place in the last eight, as he seeks to add to his sole French Open title a decade ago.
"That's the reason I came back from [knee] surgery", three-time Grand Slam champion Wawrinka said after the classic.
Nadal won by the same score as Federer on Sunday, but he was at least given a little bit more of a test by another unseeded Argentine, Juan Ignacio Londero. At 37, he is the oldest man to make the quarter-finals of a Slam since Jimmy Connors at the 1991 US Open. But I am super happy with my performance.
"It's fabulous that I can spend this time in Paris", said the 20-time Grand Slam title victor, who last played the tournament in 2015 when he lost in the quarter-finals to Wawrinka.
Tsitsipas, 20, looked on the brink of victory at 5-5 in the fifth set but could not convert any of the three break points that came his way and 28th seed Wawrinka went on to claim a 7-6 (8-6) 5-7 6-4 3-6 8-6 victory.
That was also the last year Federer played at Roland Garros, before taking the decision to skip clay entirely until returning to the surface this year.
"It's fabulous that I can spend this time in Paris", said Federer who last played the tournament in 2015 when he made the quarter-finals - where he lost to Wawrinka.
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Londero had a break point early in the third set but Nadal stamped out that danger and broke in the very next game to close in on victory.
Romanian third-seed and reigning champion in Paris Simona Halep leads the charge in the other side of the draw, along with Australian Ash Barty - both of whom play their fourth-round ties on Monday.
Federer is chasing a record-extending 21st Grand Slam title.
Stephens was the runner-up at Roland Garros past year, losing to Simona Halep.
Elsewhere, Petra Martic followed up her win over second-seeded Karolina Pliskova by rallying past Kaia Kanepi 5-7, 6-2, 6-4 to reach her first Grand Slam quarterfinal. The 19-year-old Vondrousova beat 12th-seeded Anastasija Sevastova of Latvia 6-2, 6-0 in just 59 minutes.