Anderson made the first break of the match in the third game but was pegged back by Giraldo instantly before making the decisive break three games later to go 4-3 in front.
From there, the South African was assured on serve and clinched the opener 6-4 at the second time of asking.
In the second set there was not a single break by either man and it would have to be decided in a tie-break.
Anderson had two match points at 6-4, but Giraldo saved them both before opening up the chance to level the match at one set apiece.
He couldn't take it, though, and eventually Anderson came through 10-8 in the breaker to reach round three in the French capital.