Murray overcame cramps, erratic serving, unforced errors and blustery conditions to defeat Russian Bogomolov 6-2 6-4 6-1 before admitting a possible player strike of the 2013 Australian Open would be damaging to the sport.
Murray recorded his victory despite landing only 49 per cent of his first serves and being broken on four occasions in a muddled start to his quest for a breakthrough Major championship.
The Olympic gold medallist appeared irritable in his first outing at Flushing Meadows and made 31 unforced errors in a performance he rated "a six or seven out of ten."
"I didn't serve very well but I only lost seven games in three sets so I must have done something well," Murray said.
"I played fairly well from the back of the court. I just would have liked to have served a bit better.
"A little bit of cramp - it was very, very humid on the court today. Maybe I didn't take enough fluid. I haven't played that many matches in that sort of humidity for a while so it's probably a bit of a shock to the body."
Murray said players were united in their push for an increased percentage of revenue from major championships but hoped a strike would fail to materialise.
"Who knows what is going to happen. I hope it doesn't come down to that," Murray said. "That's bad for everybody, really."
Strike action was first proposed at a heated player meeting before the Australian Open in January that led to a public rift between Rafael Nadal and world number one Roger Federer.
Nadal, an 11-time Grand Slam champion, subsequently quit his role on the players council.
"When we went through the player meeting at the Aussie Open, it was pretty brutal," Murray said. "Everyone was speaking up. The whole tour was kind of together. They still are.
"The majority of players want to see a change in the Grand Slams."
Murray declined to discuss the likelihood of a strike.
"Have no idea," he said. "There's so many things that go into something like that with lawyers, forming unions, all sorts of different scenarios that need to be thought through first.
"Right now it's a long way away, but I don't know how serious everybody is about it. If in the next month or two months they get everything sorted and ready to go then I'll have a better answer at that time."
Bogomolov failed to win his serve in the first set and blew a chance to lead by two breaks in the second set.
Murray, who lost this year's Wimbledon final to Federer, faces Croatian Ivan Dodig in the second round as he attempts to convert the momentum of Olympic success into a first triumph in a major.
"I knew after that match (the Olympic final) that everything you've gone through as a player was worth it," he said.
"I've had many tough losses. I've played tennis properly since I moved over to Spain when I was 15, so about ten years I've been playing. I've had a lot of doubts after losing.
"Even after the Wimbledon final, you have a lot of doubts about yourself. But after winning a match like that you kind of forget about all those things. It's definitely, definitely worth all the hard work."
In the women's draw, it was a mixed bag for British hopes on the opening day of action in New York.
Laura Robson, who won a silver medal in the mixed doubles with Murray at the London Olympics, progressed to the second round with a 6-3 7-6(6) win over American Samantha Crawford.
The 18-year-old secured a 6-3 7-6 (6) victory over the qualifier to set up a meeting with three-times US Open champion Kim Clijsters.
Robson was comfortable in the opening set but dropped to 0-3 early in the second before finally rediscovering her touch to take the set to a tie-break, which she won to six.
But it was a day to forget for Heather Watson, who lost 6-2 6-3 to Li Na.
Chinese star Li came into the tournament on the back of an impressive victory in Cincinnati, and was a class above the 20-year-old, ranked 71 in the world.
The ninth seed shared the first games with her opponent, even having to save a break point, before reeling off five games in a row to win the first set.
Watson, the former US Open junior champion, responded with an early break in the second to lead 3-1, but again the Chinese roared back, helped by a consistent first serve.
The match was all but won when rain intervened with Li 5-3 ahead in the second set, and after a lengthy delay, it was a brief return to court for Watson for her exit to be confirmed.
She lost four of five points on her serve following the two-and-a-half-hour rain delay, punching a backhand into the net on match point.