The 21-year-old, who made 17 double faults when she lost to fellow American Jessica Pegula in her first game in Group Steffi Graf on Sunday, committed only three double faults as she dominated the eighth seed.
Paolini is out as she had already lost to Aryna Sabalenka.
Later on Tuesday Sabalenka beat Pegula 6-4 2-6 6-3 but despite two wins has not guaranteed qualification.
"I knew today's win was important to keep myself in the tournament. If I lost, I would have been out," world No.3 Gauff said.
"I'm really happy with how I played, definitely a turnaround from my first match.
"It's the beauty of this tournament, you have another chance to prove yourself. Hopefully I can do well in my next one (against Sabalenka on Thursday) to give myself a chance to stay in this tournament."
The American took barely 10 minutes to build a 3-0 lead in the first set before Paolini went through a nine-minute battle to fend off three break points and win her first game.
Paolini got a break but Gauff drew the Italian into backhand rallies and won points with precise cross-court shots to claim the next game and take a 5-3 lead before holding to win the set.
The French Open champion's strategy of making Paolini run constantly from one side of court to the other paid off again when Gauff got back-to-back breaks to go 5-2 up in the second.
She secured victory with a powerful serve that Paolini could only hit into the net.
"I thought I served smart... I don't think Jasmine was a 100% today," Gauff added.
In the second match Sabalenka got the first break with a chip across the court to go 4-2 up, but Pegula soon broke back to make it 5-4. Pegula saved three set points in the next game, but a persistent Sabalenka clinched a decisive break with a powerful forehand from the baseline.
The US Open champion grew frustrated as she made 13 unforced errors helping the American build a 5-2 lead before saving two break points to secure the second set.
Sabalenka struggled with double faults as Pegula broke early in the third set. However, the Belarusian responded with returns near the sidelines to get two straight breaks, building a 4-2 lead that proved unassailable for Pegula.
(Pegula) is an incredible player. She always pushed me to the limit... I was thinking, I'm gonna go after my shots, stay aggressive, put all of that speed back on her and hope that I get my chance," Sabalenka said.
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