Giron clawed past former World No. 7 Richard Gasquet 7-6(5), 6-7(6), 7-5 to triumph in his first match of the year. The American saved two of the three break points he faced to advance.
"That was a battle. That's one way to start the year. I would have liked to have won in straights, I was up a mini break there in the second," Giron said. "But Richard is a phenomenal player. He's a great player, so the match is never over until it's over. He had match points in the third. I was just happy I was able to give myself a chance to get the win.
"I've been working hard in December, so it was a battle, but I'm looking forward to keep moving forward."
Gasquet served for the match at 5-3 in the decider and led 40/15, but he missed a forehand drop shot into the net and then missed his trademark one-handed backhand wide to let slip his opportunities. The 29-year-old Giron will next face second seed Felix Auger-Aliassime or Australian qualifier Alexei Popyrin.
Giron's countryman and another former college tennis player at UCLA, Mackenzie McDonald also moved into the second round. McDonald led Colombian Daniel Elahi Galan 6-3 when Galan retired due to an abdominal injury.