It was the second straight time that Medvedev started a contest one day and finished the next, having gone four sets with Australian Christopher O'Connell in a late-night battle on Thursday that wrapped early Friday morning.
"Tough, I just want to go sleep, that's it, nothing more," Medvedev told the crowd that stayed to watch the match through to its conclusion.
In a match that got underway just before 23:00 local time, Medvedev went right to work breaking Baez twice to jump out to a 4-1 lead on the way to taking the opening set.
Playing as if he was trying to end the contest before the clock struck midnight, Medvedev broke Baez again to open the second, racing to a 2-0 lead.
But Baez was not interested in a quick finish, building a 5-2 advantage in the third when the contest was stopped to allow the roof to be closed as rain began to fall.
When play resumed Medvedev, who had been growing increasingly agitated, came out with new focus, breaking the Argentine to get back on serve 5-4 and forcing the set to tie-break he would win 8-6.
"It felt like we were both playing well in the first two sets and I was just a little better on the important points," said Medvedev.
"Everything was going my way and in the third set he raised his level just a little bit, and it was enough to make the match even tougher. Lucky and good for me to stay in the third set and not finish at four."
The win set up Medvedev for an intriguing last 16 meeting with Australian danger man Alex de Minaur.
Medvedev and de Minaur have met six times with the Russian winning four but the Australian has won the last two, including a quarter-final clash at last month's Canadian Open.
Medvedev couldn't be joined by compatriot Ekaterina Alexandrova in the next round, with Wimbledon champion Marketa Vondrousova defeating her 6-2, 6-1 to storm into the fourth round at Flushing Meadows for the first time in five years.
The Czech ninth seed, who made her Grand Slam breakthrough at the All England Club last month, has flown under the radar in New York and came into the clash with her Russian opponent having dropped only seven games.
The 24-year-old surrendered her serve twice in the first set but showed all her artistry at Louis Armstrong Stadium to break Alexandrova four times and wrap it up easily.
Alexandrova continued to look shaky on her delivery in the next set and allowed Vondrousova to surge ahead 4-0 with a third double fault of the contest before finally producing a hold two games later to avoid a bagel.
But there would be no late comeback for the 22nd seed, with Vondrousova completing the victory in only 56 minutes when Alexandrova fired a backhand wide.
"I felt really good," said Vondrousova, who will meet American hope Peyton Stearns in the next round.
"I was waiting the whole day, I didn't know what to expect. It's actually my first night session here and I really enjoyed it. Thank you so much.
"I'm very happy with my game, it's a lot of pressure and I didn't know how it's going to work. I'm just grateful to be here and play healthy again. I'm enjoying my time in New York."