The third seed began slowly and lost his opening two service games but he had no trouble breaking the Bogomolov serve, doing so nine times on the way to a 6-2 6-4 6-1 victory.
Murray was reasonably satisfied with his performance, saying: “I thought it was a six or seven out of 10. I didn’t serve particularly well at the beginning but I won in straight sets against a tough player, and that’s what I needed to do.”
The Scot was hoping to carry the momentum from his golden Olympics into the year’s final grand slam but he could not have made a worse start, dropping his opening service game with a series of errors.
There had been a rain delay of more than two hours as a heavy thunderstorm hit Flushing Meadows and Murray appeared to be mentally still in the locker room.
He pulled level with an immediate break of the Bogomolov serve only to be broken again as the strange start to the match continued, the Scot bouncing his racquet on the court in frustration.
Bogomolov, who switched allegiance from the US to Russia last December, was the opponent for one of the lowest moments of Murray’s career in Miami last year when, in a post-Australian Open final slump, he lost a fourth straight match.
The world number four had won their last two meetings, though, and managed to get himself back on terms once more with yet another break for 2-2.
And this time he finally held on to his own serve, although not without saving two more break points, and at last began to look more comfortable.
He took his third chance to gain a third straight break of the Bogomolov serve with a lovely angled backhand, and then ensured his opponent ended the set without holding serve at all.
Although he had improved markedly, all was still not well with Murray, who was getting little more than 30% of his first serves in and muttering to himself.
He promptly dropped serve again at the start of the second set and this time he could not retrieve the situation immediately as Bogomolov finally held serve.
The 29-year-old has had a disappointing season, dropping from 34th in the world at the start of the year to 73rd, but he was playing aggressively and really taking the game to Murray.
The Scot had to save two more break points to avoid going 4-1 behind, but he held on and then drew level at 4-4 when Bogomolov drilled a forehand long.
And that proved to be the turning point as Murray broke again to win the set, powering a forehand winner out of the reach of his opponent.
It had been a strange contest, as Murray’s grand slam openers often are, and the ups and downs continued at the start of the third set as the Scot broke serve and was then promptly broken straight back.
He has a habit of conceding his serve at such moments, and, after making it three breaks in a row, he almost succumbed to a fourth but this time managed to stave off the threat.
That was the last throw of the dice as far as Bogomolov was concerned and Murray, who had begun to struggle a little with cramp in the humid conditions, broke again before clinching victory with a backhand winner after two hours and 15 minutes.
He looked relieved rather than pleased as he walked to the net to shake hands and admitted afterwards he had found the match physically challenging.
He said: “I was struggling a little bit, there were a lot of long games, long points, we both did a lot of running.”
Meanwhile Sam Stosur won the first 19 points as she began the defence of her US Open title in style with a 6-1 6-1 victory over Croatia’s Petra Martic.
The Australian, seeded seventh this year, was a surprise winner over Serena Williams 12 months ago but is under the radar again this time after failing to build on her maiden grand slam title.
She certainly made a statement today, though, leading 21-year-old Croatian Martic 4-0 and 40-0 before she dropped a point, and she was hardly less dominant thereafter.
It took Stosur only 51 minutes to clinch victory and set up a second-round match against either Edina Gallovits-Hall of Romania or Switzerland’s Stefanie Voegele.