Good guess with the Jennings idea. He pitched a 5-hit shutout in his debut at Shea back in Aug 2001. Also hit a homer and pair of singles in the game for one of the best all-around debuts by a pitcher ever.
Atlee Hammaker pitched four shutout innings of relief in his debut in Aug 1981 when he came up through the Royals system.
Dave Dravecky also pitched in relief when he made his debut with the Padres in June 1982.
Of the two pitchers who serve as the answers to this question, one is in the Hall of Fame.
Well we're talking about Texas and we're talking about low-hit games so I have to toss Nolan Ryan out there as a guess for the Hall-of-Famer (don't get me started on Ryan in the HOF).
The Dominican Dandy made his MLB debut for the Giants on July 19, 1960 at The Stick in San Fran. Marichal went the full nine innings, allowing one hit and a walk to go with 12 strikeouts. The lone hit was by Clay Dalrymple, a 2-out pinch-hit single in the eighth.
The only other pitcher to accomplish this feat in the NL was James C. 'Jimmy' Jones. The Dallas native made his debut for the Padres and 1-hit the Astros in The Dome on Sep 21, 1986. The lone hit was a 2-out triple in the third inning by Houston starting pitcher Bob Knepper. Jones went the full nine innings allowing no walks and striking out five.