The second seed was out of sorts at the ATP 1000 event on Wednesday as he struggled to stay in rallies with the world number 25, but he eventually progressed 7-5 3-6 7-6 (9-7) after saving five match points.
Federer was behind in all three sets of the match at some stage, but showed his character to win four games in a row to take the first and then to battle in a decisive tie-break.
After Mayer had levelled the match he looked on course for a surprise win over the Swiss, but he fluffed his lines when the first two match points arrived.
Federer's battle was not to end there, though, as Mayer had a further three chances to win in the tie-break before the world number two found some momentum to win three successive points.
"[It] could be my greatest escape thus far," said the 17-time Grand Slam winner.
"I've had some good ones over the years, but this one might just top it all because I was literally out of the point at one stage.
"I had to come up with some shots that were hard to do anyways. But on match point even more so.
"So I feel a bit sorry for Leonardo, I must say, but that's how it goes. You win some, you lose some."
Federer will now face Spaniard Roberto Bautista Agut in the third round after he beat Vasek Pospisil 7-6 (7-1) 3-6 6-4.