Despite not dropping a set against the world number 73, Murray was visibly disappointed with his performance at times on the Arthur Ashe Court, particularly with his inability to consistently hold his serve.
The Brit was actually broken in his first two service games of the match as he struggled to find any pace on his second serve.
Bogomolov was having similar problems though and as a result the match was tied at two games apiece.
It was Murray who eventually made the breakthrough by winning his service game for a 3-2 lead, before going on to claim the next three games to wrap up the first set 6-3.
He failed to build upon that success immediatley though, with Bogomolov securing a break of Murray's serve at the first time of asking. The Russian went on to open up a 4-2 advantage as Murray sought to level up the score.
However, a combination of unforced errors from the 29-year-old and a series of forehand winners from Murray allowed the Scot to win four games in succession, which in turn saw him take the second set 6-4.
Olympic gold medallist Murray appeared to be heading for a straightforward triumph in the third set as he took a two-game lead over his opponent ranked 67 places below him in the standings.
Despite being broken once again by Bogomolov and an attack of cramp in his hamstring, the 25-year-old held out to win the set 6-1 and in doing so secured a spot in the next round, where he will take on either Ivan Dodig or Hiroki Moriya.