He is on course to break into the top 10 for the first time next week and was ultimately too strong for talented Frenchman Gaston, winning 6-3 7-5.
Norrie looked set to win more comfortably, recovering from dropping serve in the opening game of the match to move a set and 5-3 ahead.
He held five match points on his opponent’s serve and then another on his own in the following game but was unable to take any of them, with Gaston using all his trickery to claw his way back into the match.
However, Norrie broke serve again to move 6-5 in front – with Gaston smacking the net and then a ball into the crowd in frustration – and finally clinched an eighth match point.
He told Amazon Prime Video: “It was one of those days where it was tough to close it out. I was a little bit stressed there. I managed to sneak through. I just got a little bit tentative and lost my feet and wasn’t moving enough.
“I was really happy with how I responded to getting broken. He’s a tricky guy. You feel like you need to be really aggressive otherwise he’s going to throw in a drop shot at any moment. I think that’s something I need to work on.”