Murray, playing in his first tournament since skipping Wimbledon through injury, converted on his seventh match point to beat Mackenzie McDonald in three sets.
The two-time Wimbledon champion toiled for more than two-and-a-half hours against world number 80 McDonald, eventually taking the match 3-6 6-4 7-5.
McDonald, 23, broke two of Murray's opening three service games and wrapped up the first set 6-3 inside 40 minutes.
But Murray drew things level, taking a crucial game nine in the second for a 5-4 lead and serving out the set.
A 73-minute deciding set followed, in which Murray took an early break and found himself 5-4 up and serving for the match.
But McDonald saved five match points and converted his second break point to draw things level at 5-5.
Murray broke back immediately for a 6-5 lead following a controversial call when McDonald's racket was adjudged to have crossed the plane of the net at 30-30. The former British number one then made no mistake in serving out the match.
The 31-year-old skipped Wimbledon earlier this year and is playing in Washington as he continues his protracted recovery from long-term hip trouble.
Murray had surgery on his right hip in January, only returning to action on June 18, facing Nick Kyrgios at Queen's.
He later overcame Stan Wawrinka at Eastbourne, only to lose in straight sets to Edmund two days later.
Murray has signed up to play in the China Open at the beginning of October.
The three-time grand slam champion continues to add more tournaments to his schedule as he scales up his return from hip surgery.
The Scot will play the Masters events in Toronto and Cincinnati prior to the US Open and is set to begin his Asian swing at the China Open beginning on October 1.
Murray won the title in Beijing on his last visit in 2016, part of a run of five successive tournament victories that propelled him to world number one.
Related tennis links
