The Briton got the better of her 23-year-old American opponent in two hours and three minutes in the final round of qualifying for the main draw.
The first set was closely fought with Watson edging 5-3 ahead at one stage, but then missed out on two set points with unforced errors to lose the game.
However, the 21-year-old made no mistake in winning the following game, although she struggled to make it easy for herself in the second set.
Watson was 5-4 up in the second at one stage and forced two match points against Falconi, but squandered the chance to make it a straightforward victory.
Falconi threatened to take the contest into a third and deciding set, but Watson won the second set tie-break 7-1.