French Open champ Gauff, who is bidding to become the first woman to win back-to-back titles at the WTA 1000 event, had battled through three-setters in the previous two rounds and had to overcome stiff resistance from the 66th-ranked German on Thursday.
"I'm happy with how I played today. She's a tough opponent, she hit a couple of great shots on the run," Gauff said.
"I think I need to stay confident in my game and not be too passive when I have the lead. I played one passive point in this match, but otherwise I played well."
It was a fast and furious start to the first set as both players fired off a string of winners and traded early breaks as the momentum swung wildly.
Following a run of five straight breaks of serve it was defending champion Gauff who finally seized control, taking a 5-3 lead when Lys sent a backhand wide and then consolidating to clinch the first set.
The world No.3 appeared to have found her range on serve in the second set and eased through a couple of holds, but a brief wobble and a few double-faults from the American added some late drama.
Serving for the match at 5-4, however, Gauff held her nerve to close out the win in an hour and 28 minutes.
World No.4 Anisimova, who was playing with a bandage on her ankle after pulling out of last month's Korea Open due to injury, was tested by Paolini in a battle that lasted two hours and 47 minutes.
A break in the seventh game handed Anisimova a chance to win the first set, but world No.8 Paolini fought back from 5-3 down to force a tiebreak, which she won to take the lead in the contest.
But Anisimova battled back in the second, taking a 4-0 lead which was enough to secure the set despite a late fightback by Paolini.
The Italian was on the verge of taking a crucial 5-3 lead in the third, but Anisimova saved six break points in a marathon game to hold serve.
Anisimova won the next two games to wrap up victory, reaching the semis and also qualifying for the WTA Finals, which are set for next month in Saudi Arabia, for the first time.
"It was a goal that I set for myself at the beginning of the year. I think it was one of the only goals I made," Anisimova said.
"I'm excited to play Coco... it's going to be fun to play another American here."Â
