The British number four, who is attempting to qualify for his first Grand Slam tournament, let a match point slip by before prevailing 6-7 7-6 6-4 against the up-and-coming American.
After coming through a tight first set, Broady moved to the brink of victory when creating a match point late in the second, but after Donaldson struck back, he overturned a 5-3 deficit in the tie-break to take the match into a decider.
The 18-year-old had the momentum but Broady provided an immediate response, breaking in the opening game of the third before having the chance to serve for the match at 5-4.
Donaldson created an opening to extend the contest at 30-40, but Broady hit back with a volley and two aces to record one of the biggest wins of his career.
The left-hander will now face the experienced Michael Russell in the final qualifying round.