The rookie also got a cool birthday gift in the process.
"He has great poise," Rangers manager Ron Washington said of his young starter. "He's not afraid to lose any of his pitches in any count. He attacked the strike zone. The most important thing is they didn't know what he was going to throw."
Hunter (1-1) was making his sixth career start and third this season. The right-hander allowed three hits and struck out five in 5 1/3 innings to help slow down baseball's top-scoring offense.
Hunter went 0-2 with a 16.36 ERA in three starts last season. In the minors, he developed a changeup and a cut fastball.
Along with his first victory, Hunter's sister had a baby Friday to make him an uncle for the first time.
"It's fun to be able to use four pitches for strikes," Hunter said. "It keeps them off-balance. They had a lot of broken bats tonight. It went my way."
Jason Jennings, C.J. Wilson and Frank Francisco pitched 3 2/3 innings of scoreless relief to complete the three-hitter, which equals the fewest hits Tampa Bay has had this season. Francisco pitched a perfect ninth for his 13th save.
"We did not have a good game," Rays manager Joe Maddon said. "We made some mistakes and didn't come up with the clutch hit."
Blalock, who hit a two-run homer in the ninth to beat the Los Angeles Angels on Wednesday night, had a two-run shot in the fourth.
B.J. Upton had an RBI single in the fifth, and the Rays were threatening to score more with runners on second and third in the inning.
"They've got my back," Hunter said. "It's fun to play for a team like that."
Scott Kazmir (4-5) made his second start for Tampa Bay, which has won eight of 11, since coming off the disabled list June 27. The left-hander gave up three runs -- one earned -- and struck out six in five innings.
Blalock was 0 for 7 lifetime with three strikeouts against Kazmir before he drove a 2-2 pitch into the seats in right to put the Rangers up 2-0.
"He made me pay," Kazmir said. "You don't throw a hanging slider to Blalock."
Texas scored its second unearned run in the fifth.
Kinsler reached on a fielder's choice and took second when Ben Zobrist made a wild throw on a double-play attempt.
Marlon Byrd brought Kinsler home with an RBI double that made it 3-1.
Tampa Bay has moved RHP Jeff Niemann, who was Sunday's scheduled starter, into the bullpen for the weekend series. Matt Garza will start Sunday against the Rangers. Maddon said Niemann will likely be back in the rotation next week. ... Washington said RHP Vicente Padilla (sore shoulder) is scheduled to start Tuesday against the Angels. Padilla was originally slated to pitch Saturday, but was pushed back. ... There was no delay when a bank of lights along the third-base line went out in the eighth inning. All lights were working again a few minutes later. On June 13, the Rangers game against the Dodgers was delayed nearly two hours when a light standard went dark.