The Bristol-born youngster showed signs of promise against Hungarian Babos, who secured a straight sets victory (6-2, 6-3) in just under an hour.
Swan was on the end of a first set masterclass from Babos as the doubles specialist raced into a 5-1 lead before seeing out the set in 28 minutes.
Former Bristol Grammar pupil Swan retained her composure to hold serve in the first game of the second set and almost broke her opponent only for Babos to end an exciting rally to save break point to level.
The 17-year-old, who was the youngest player to play on the senior stage in 2016 after receiving a wildcard entry, began to showcase why she is highly-regarded as Britain’s next top tennis star.
Her powerful serve produced six aces during the tie and helped her take a 3-2 lead but Babos eventually turned the screw and sent Swan chasing frantically around the court, on the way to a comfortable win.