"The beginning, the first match, is always a little bit new even if I've played here plenty of times", said Nadal, who meets Aussie Thanasi Kokkinakis next.
Nadal, who is undefeated in U.S. Open first rounds, may have had reason to worry about Millman, who upset Roger Federer previous year in the fourth round at Flushing Meadows.
After his Russian opponent broke to lead 4-3 in the final set, Tsitsipas, who had already received a coaching warning and time violation, did not take kindly to being told to speed things up by French chair umpire Damien Dumusois.
Two-time French Open runner-up Thiem was upset by Italian giant-killer Thomas Fabbiano 6-4, 3-6, 6-3, 6-2, the Austrian fourth seed slumping to another first-round exit, having lost at the same stage at Wimbledon.
'I'm very happy with the victory, and thank you very much to everybody for the support'.
Greek eighth seed Tsitsipas struggled with cramps and accused umpires of having "preferences" after he fell to a 6-4, 6-7 (5/7), 7-6 (9/7), 7-5 defeat by Next Gen rival Andrey Rublev in a gruelling four-hour slog.
"I feel like I'm doing the same routines on the court, the same execution, the same... I'm far away from 100 percent", an ailing Thiem said.
Riding a nine-match winning streak after collecting her maiden WTA title last week at The Bronx, in-form Pole Magda Linette was too hot for Sharma, winning 6-3, 6-4.
North Korea Says it is 'Ready For Dialogue or Standoff' With US
The South Korean government, for its part, made a decision to downgrade Japan's trade status. But unlike GSOMIA, Seoul and Tokyo would negotiate to exchange information "case by case".
"The chair umpire was very incorrect in what he was telling me during the match", Tsitsipas said afterward. "I don't feel like I'm chasing something".
But, the German sixth seed managed to get past Albot 6-1 6-3 3-6 4-6 6-2 after three hours, 10 minutes.
Russian ninth seed Karen Khachanov was another early casualty after he lost 4-6, 7-5, 7-5, 4-6, 6-3 to Canada's Vasek Pospisil.
Reigning women's US Open champion Naomi Osaka confronted severe nerves as she overcame Russia's Anna Blinkova, the world No. 84.
"But I think nearly everybody goes through that".
"You don't like tennis when you're travelling around the world and you're sacrificing week for week and you feel like the game's not there and you're getting pushed out early in Slams and also tournaments, you feel like all you want to do is run in the other way and just not be there", Federer said.
World number 43 Rublev, who beat Roger Federer in Cincinnati earlier this month, said: "It was tough conditions for both of us and we played an fantastic match".
"I actually thought that made my game better".