Djokovic, winner of four of the past five Slam titles and 16 in all, was treated for a sore left shoulder throughout his 6-4, 7-6 (7/3), 6-1 victory over Argentina’s 56th-ranked Juan Ignacio Londero.
Djokovic next faces 27th-seeded Serbian compatriot Dusan Lajovic or American Denis Kudla, but the injury casts grave doubts on his bid to be the first US Open repeat winner since Federer won from 2004-2008.
Federer, a 20-time Grand Slam winner seeking his first US Open crown since 2008, rallied to beat 99th-ranked Bosnian Damir Dzumhur 3-6, 6-2, 6-3, 6-4.
The 38-year-old Swiss third seed, who also dropped the first set against Indian qualifier Sumit Nagal in his opener, answered the wake-up call after conceding the first set with 17 unforced errors.
Japanese seventh seed Kei Nishikori, the 2014 US Open runner-up, beat 108th-ranked American Bradley Klahn 6-2, 4-6, 6-3, 7-5.
Only 10 of 32 singles matches were played as rain wiped out play except in enclosed stadiums, with Russian fifth seed Daniil Medvedev and three-time Grand Slam champion Stan Wawrinka among those postponed to Thursday.
French Open champion Ashleigh Barty, the second seed from Australia, eliminated 73rd-ranked American Lauren Davis 6-2, 7-6 (7/2), saving a set point before dominating the tie-breaker.
Czech third seed Karolina Pliskova, seeking her first Grand Slam title, ousted 202nd-ranked Georgian qualifier Mariam Bolkvadze 6-1, 6-4.
Ukraine fifth seed Elina Svitolina defeated seven-time Grand Slam winner Venus Williams 6-4, 6-4 and US 10th seed Madison Keys, the 2017 US Open runner-up, trounced China’s Zhu Lin 6-4, 6-1. (AFP)