Wu broke Chardy in his first service game after the Frenchman committed two double faults.
Federer had won all four of his previous matches against Dolgopolov and his perfect record against the Ukrainian was never really under threat.
After wasting three break chances, Federer eventually converted one to move ahead, only for Schwartzman to hit straight back - the Swiss uncharacteristically firing a forehand long.
It's easy to forget that sports stars have family lives to maintain while trying to stay at the pinnacle of their discipline, but making time for his children despite a busy schedule is nothing less than we'd expect from one of the classiest guys in sport.
'Was a very quick match in all aspects', said Nadal, who has not been losing any match since Cincinnati quarter-finals to Kyrgios. 'I think Jared started playing very well at the beginning, serving big, and then hitting very strong from the baseline.
"If that happens, it will be, I think, a very important achievement for me", said Nadal.
Meanwhile, Juan Martin del Potro needed all three sets to reach the second round, beating qualifier Nikoloz Basilashvili 6-2, 3-6, 6-4.
A 696th hard-court victory was secured after 93 minutes, the Australian Open and Wimbledon champion setting up a meeting with Alexandr Dolgopolov in a 7-6 (7-4) 6-4 win. You can tell that there was a lot of effort put into it. "This is something that is either going to happen or not going to happen", he said. I just had to be ready for anything.
Some of the players who aren't just willing to attack but also serve-and-volley are Feliciano Lopez, Sergiy Stakhovsky, Vasek Pospisil, Nicolas Mahut and Dustin Brown. The set went to a tie-break, and despite having a sloppy start, the 36-year-old tennis icon made the right adjustments to clinch the set. During the tiebreaker, he was docked a point for inappropriate language.
Two upsets of note, Viktor Troicki edged the reeling Dominic Thiem 6-3, 3-6, 7-6 (5). "I was even a bit surprised clinching match point".
Tomorrow, Simon faces countryman Richard Gasquet and also Marin Cilic meets Steve Johnson.
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