Murray kicked things off with his first straight-sets victory at a grand slam for five years, defeating Argentinian 24th seed Francisco Cerundolo 7-5 6-3 6-3.
Having gone into the contest short on wins and with concerns over a cramping problem that has affected him in several matches, this was a very positive start to his campaign.
Murray said: "Mentally it's not easy going into a five-set match in those conditions after the way the last few weeks have been for me, and I did well.
"I did well physically and mentally to put that to the back of my mind and find a way to get through. So I was really happy with that.
"It was really tricky conditions out there. At the beginning it was actually OK, but towards the end it was really getting pretty hot and humid. I thought I did a pretty good job out there. Could have closed some of the sets quicker, but overall it was a solid performance."
Murray will next take on young American Emilio Nava as he bids to reach the third round at Flushing Meadows for the first time since 2016.
Although Draper's clash with Finn Emil Ruusuvuori was his first senior match in New York, such has been his meteoric rise this season that he was already viewed as someone to avoid in the draw.
And that will not change after an efficient 6-4 6-3 6-3 success to set up an intriguing second-round clash against sixth seed Felix Auger-Aliassime.
"It's my third grand slam, second on merit," said 20-year-old Draper. "I'd say it's a really important win for me. He had me at Queen's, he played an incredible match there, so there were definitely doubts in my mind before the match.
"But I was proud of the way I handled my first time here, thought I played a solid match. I think I can still improve tennis wise but mentally and physically it was a good performance."
The result of the day came from Dart, who stunned 10th seed Daria Kasatkina 7-6 (8) 1-6 6-3 for her first victory over a top-10 player.
The British number two had never previously won a main draw match at Flushing Meadows and looked to have a very tough encounter on her hands against Russian Kasatkina, who is ranked ninth after winning a WTA Tour title on Saturday.
But Dart battled superbly in hot conditions on Court 12, recovering from a shaky second set and then coming from behind in the third to continue her career-best year.
The 26-year-old's reward is a very winnable second-round match against Hungary's Dalma Galfi.
The only British defeat was an expected one as Kyle Edmund, playing his first grand slam singles match for two years following three knee operations, fell to a 6-3 7-5 6-2 loss against fifth seed Casper Ruud.
Defending champion Emma Raducanu, seventh seed Cameron Norrie and 20th seed Dan Evans will all bid to join the British winners on Tuesday.