Sitak and Monroe went down an early break in the second at 1-2 before digging deep to force a decider.
"It didn't look like it was going to go well for us, but we stayed strong every point and broke back right away and in the tie break we had some really cool points with reflex volleys and good tennis in the tie break to win it." Sitak said.
Sitak and Monroe made the perfect start to the deciding set with an immediate break of serve to take control. The Kiwi was the best player on the court making some clutch volleys and big passing shots.
They had three match points at 5-3 with Gabashvili and Schwartzman holding serve from 0-40 down before Monroe served for the match at 5-4. He eventually closed it out on his second match point with Sitak putting away a volley at the net.
"That game at love 40 we played all the points well but they came up with some big shots. Nick (Monroe) was serving for the match at 15-30 it was a crazy point, there was a let cord over me and Nick got it and then Gabashvili missed a volley before Monroe served out the set."
Sitak and Monroe face a tough second round encounter with 9th seeds Poland's Lukas Kubot and Austrian Alexander Peya.
"Tomorrow will be a different match up against doubles specialists but the good thing is they have just started playing together so we might have an edge. But it will come down to execution, who serves better and takes control at the net." Sitak said.
Meanwhile Kiwi Marcus Daniell and American Brian Baker will face 11th seeds Henri Kontinen from Finland and Australian John Peers in their second round match late tonight.
Matt Brown is at the French Open thanks to Emirates Airline