Federer reported a right leg injury ahead of his match against Stefanos Tsitsipas, while Osaka said her right hand was injured before she was to play Kiki Bertens.
Federer saved all seven break points he faced to beat Joao Sousa 6-4, 6-3, then was to play Borna Coric for a spot in the quarterfinals.
Osaka easily won the first set before Vekic looked to have seized control of the match by winning the second and building up a 5-1 lead in the third.
Novak Djokovic lost 6-4, 6-4 to Philipp Kohlschreiber in the third-round while Naomi Osaka lost to Belinda Bencic 6-3, 6-1 in the fourth round.
The move comes two and a half weeks after Osaka surprisingly split from coach Sascha Bajin, who previously worked with Serena Williams.
The two-time major champion had trouble with her serve and was beaten 6-3, 6-3 by 67th-ranked Kristina Mladenovic at the Dubai Championships.
The two-time Grand Slam champion admitted that the upcoming tournaments, including Dubai Championship, are important for her.
Osaka's second consecutive major title, following her success at the U.S. Open last year, helped her rise three spots.
Novak Djokovic, a 31-year-old already ranked No. 1, now has won the past three men's Grand Slam titles.
The victory allows the 21-year-old Osaka to become the first tennis player from Japan to reach No. 1 in the rankings.
Petra Kvitova defeated Danielle Collins 7-6 (2), 6-0 while Naomi Osaka downed Karolina Pliskova 6-2, 4-6, 6-4 in the semi-finals to set up a summit clash against each other.
Pliskova broke Osaka for a second time to go up 5-4 in the second set and then hammered her sixth ace to secure her 11th career title.
Osaka hammered nine aces over the course of the victory and dropped just eight points in eight service games over the course of the match.
Serbia's Novak Djokovic jumped three spots to third after he captured his third US Open title and 14th overall grand slam.
Williams had lost the final 6-2, 6-4 to Japan's Naomi Osaka but the game made the headlines for her allegation that Ramos treated her differently than he would a male player.
For the first time in the United States Open, the trophy ceremony was drowned out by whistles and boos in support of the loser and against the champion.
Williams was cited by official Carlos Ramos for three code violations during her 6-2, 6-4 loss to Naomi Osaka in US Open finals.
Osaka grew up rooting for Serena Williams, even did a report on her way back in third grade. Her dream was to play Serena at the U.S. Open.
On Saturday, Williams will face No. 20 seed Osaka, a 20-year-old who is the first Japanese woman to reach a Grand Slam final.
Both players won, Osaka decisively and Nishikori narrowly to give Japan a men's and women's semifinalist at the same Grand Slam tournament for the first time.
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