Profar hit a tiebreaking RBI double in the 10th inning, helping the Rangers beat the Tampa Bay Rays 4-2 on Saturday night.
"He's not afraid of the game," Texas manager Ron Washington said. "At 19 years old, that's special."
It was only Profar's second career start and third game overall. In his major league debut and first start last Saturday at Cleveland, Profar went 2 for 4, including a homer in his first plate appearance.
"A good first two games," Profar said with a smile.
Geovany Soto had a two-out double to deep center off Kyle Farnsworth (1-4). Profar then made it 3-2 when he drove in pinch-runner Leonys Martin on his hit past a diving Carlos Pena down the first-base line.
"As soon as Geno got that double, I was thinking we've got them," said Profar, whose parents were at the game.
Texas took a 4-2 lead when Josh Hamilton, who earlier hit his 40th homer this season, was hit by a pitch with the bases loaded.
Pena was called out looking to end the game with a runner on second and was ejected for arguing the call by plate umpire Mike Estabrook.
"Their call stands," Pena said. "Was it a strike? No."
Tampa Bay tied it 2-all in the eighth when pinch-runner Rich Thompson scored from second after second baseman Ian Kinsler, positioned in shallow right field as part of a defensive shift, misplayed pinch-hitter Pena's two-out grounder.
Yu Darvish gave up two runs and two hits in eight innings for Texas, which played its third consecutive extra-inning game. Darvish, with eight strikeouts, set a team rookie record for a season with 196. The old mark was 189, set by Edwin Correa in 1986.
"I was able to keep my team in the game," Archer said. "That was pretty much what our goal was."
Price was scratched from Saturday's start due to left shoulder soreness and is expected to return for a series next weekend at the New York Yankees.
Hamilton put the Rangers up 2-0 on a towering two-run shot to center off Archer in the fourth, giving him 119 RBIs.
"That's what good hitters do," Washington said. "And Hamilton is definitely one of the best in the American League and all of baseball."
Hamilton was hit by Archer's high pitch during his next at-bat in the sixth. Tampa Bay's Jose Molina stepped in front of Hamilton after the pitch, but the Rangers slugger appeared to briefly laugh with Molina before heading to first.
Both benches were warned by Estabrook.
"That was inappropriate," Rays manager Joe Maddon said. "There was no intent on our part. It was a young umpire who did not want the game to get out of control. I understand that in a sense, but there was no intent on our part."
Hamilton is the fifth player in Rangers history to hit 40 or more homers in one season, joining Juan Gonzalez (five times), Rafael Palmeiro (three times), Alex Rodriguez (three times) and Mark Teixeira (one time).
After Sam Fuld singled leading off the first, Darvish didn't allow another baserunner until Jeff Keppinger started the sixth with a single. The Rangers right-hander walked Ryan Roberts and then threw a wild pitch that moved the runners to second and third.
Molina struck out before Fuld's RBI grounder cut the Rays' deficit to 2-1. B.J. Upton followed with a fly out.
"The razor-thin line between winning and losing was on display tonight," Maddon said.
Archer got help from Roberts, who grabbed Profar's one-hop shot at second base and turned it into an inning-ending double play with runners on first and third in the fifth.
Opponents had 25 consecutive successful stolen base attempts against the Rangers, dating to Aug. 6, until Soto threw out Fuld attempting to steal second in the first ... Tampa Bay LF Desmond Jennings was out of the starting lineup due to lower back stiffness. ... Rangers C Mike Napoli (left quadriceps) is set to join Double-A Frisco for a rehab assignment. ... Texas reliever Alexi Ogando has been slowed the past few days because of biceps tightness.