Cole's dominant performance came two nights after Trevor Bauer scattered six hits and struck out 11 over seven innings in the Bruins' 11-3 first-round win over Florida.
"They compete against each other. That's what our pitching staff has done all year," UCLA coach John Savage said.
The win gives UCLA (50-14) control of Bracket 1 and three days off. The Horned Frogs (52-13) will play Florida State in an elimination game Wednesday, with the winner meeting the Bruins on Friday.
"Whenever you get into the College World Series and win the first two games, you have to be happy where you're at," Savage said. "When I say we haven't done anything, I mean no disrespect. We certainly beat good Florida and TCU teams. But this team is on a mission. We have a lot of baseball to play."
The Frogs (52-13), who came into the game hitting a CWS-best .341, scored three runs on four hits in the seventh inning off Cole (11-3).
The New York Yankees' 2009 first-round draft pick allowed Matt Curry's base hit to right leading off the second and didn't give up another until Jason Coats' hard grounder past first baseman Justin Uribe with one out in the seventh.
After Curry and Jantzen Witte followed with singles to load the bases, the pro-TCU crowd rose as Cole ran the count full against Taylor Featherston.
"My stomach dropped there for a while," Featherston said. "It probably was the most fun at-bat in my life."
Featherston fouled one off before ripping a two-out triple beyond the reach of diving center fielder Beau Amaral. The ball rolled to the wall in left-center, and the Frogs were within 5-3.
"You know it's coming," Cole said. "TCU is a good offense. You look at their numbers, and you're pretty impressed. It was just a matter of time before they put in a few good hacks."
The 6-foot-4, 220-pound Cole, who regularly touched the mid 90s with his fastball, pitched a scoreless eighth before turning the game over to closer Dan Klein, who earned his 10th save.
Cole allowed five hits, walked two and hit a batter. He has given up five or fewer hits in 13 of his 18 starts.
Savage said he almost took out Cole in the eighth, but the pitcher talked him out of it.
All Cole said to his coach: "Trust me."
After the way Bauer and Cole have pitched, the onus is on the next starter, Rob Rasmussen. The three form the core of a pitching staff that joins Texas' as the only ones in the nation with a team ERA under 3.00. Any of the three UCLA starters would be a No. 1 on almost any staff in the country.
"It's not about being the best guy," Savage said. "Gerrit has been our Friday guy all year. He really has done an unbelievable job for us. His talent is off the sheets."
The Bruins went ahead 5-0 in the third inning on Regis and Gelalich's home runs against TCU starter Kyle Winkler (12-2). Regis' two-run homer was his ninth of the season, with all coming since May 7. Two batters later, Gelalich followed with his second homer.
The Bruins, who scored in eight innings of their win over Florida on Saturday, got runs each of the first three against Winkler.
Amaral went 3 for 3 and reached base four times, with an RBI and two runs scored. He has reached on nine of his 10 plate appearances in the CWS and is batting .516 (16 for 31) in the NCAA tournament.
Winkler, who pitched 7 2-3 shutout innings against Texas in the super-regional final, was charged with five runs in 2 1-3 innings.
"Wink has had a phenomenal year for us and has really pitched well the back half of the season," TCU coach Jim Schlossnagle said. "It wasn't his day. He pitched behind in the count and walked some guys. He'll pitch again and will be better next time."
Schlossnagle said he's not concerned about how his team will respond in an elimination situation.
"One thing we talked about is you can lose two games and win a national championship," he said. "You can't lose any more between now and the championship series. We've used our Mulligan."