The Bruins (14-4-5) will play Virginia in the championship game Sunday. Virginia beat Maryland Baltimore County, 1-0, in the first semifinal.
Leo Stolz's corner kick deflected to Gasper, whose shot bounced off Providence defender Brandon Adler into the net in the 105th minute.
"I just swung away and I got a lucky deflection," Gasper said. "Good things happen when you put the ball on target, and that's what happened."
Larry Ndjock tied it for UCLA in the 81st minute with his second goal of the game.
The Bruins, who beat Virginia 2-0 in the 1997 championship game, advanced to the final for the first time since 2006. They have won four NCAA titles, the last coming in 2002.
Gasper, who is from Alexandria, Va., and committed to Virginia before switching to UCLA, will see several familiar faces in Sunday's final.
"It's a dream come true playing against them," he said.
Mac Steeves scored twice to help Providence (16-5-2) rally from a 1-0 halftime deficit.
The Friars were playing in their first College Cup after setting a school record for victories and winning their first conference championship.
"I thought we deserved more," Providence Coach Craig Stewart said. "I thought on the day we were the better team, and that's not to take anything away from UCLA."
All three of UCLA's goals came off deflections.
Ndjock rebounded his own saved shot off Providence goalkeeper Keasel Broome to score in the 43rd minute.
He tied it at 2-2 in the 81st minute after Abu Danladi's shot was blocked by Adler. The ball deflected high into the air, and Ndjock headed it past the diving Broome.
"There have been times in the season when it has gone our way," Providence's Phil Towler said. "Tonight, obviously it didn't three times."
The Bruins scored three goals for the third consecutive tournament game, but they have allowed seven goals during that span.
"Obviously, we continue to give up some goals," UCLA Coach Jorge Salcedo said. "While it's pretty entertaining for the fans, it's getting pretty difficult on me."