The Rays, bolstered by pitching depth and defensive work, captured the opener in the best-of-seven American League Championship Series with game two set for Monday.
"We take pride in our pitching and our defense and it was good for us to show that tonight," Zunino said.
The Astros are in their fourth consecutive ALCS while the Rays are playing for a World Series berth for only the second time in club history.
Houston opened the scoring on Jose Altuve's solo homer in the first inning but Tampa Bay equalized on Randy Arozarena's solo homer in the fourth inning.
The Rays plated the deciding run in the fifth when Willy Adames walked, advanced to second and third on ground outs and scored on Zunino's single to centerfield.
"With a guy in scoring position, I was just trying to get something up high in the zone," Zunino said. "I was lucky enough to get some contact on it."
Rays starting pitcher Blake Snell, a 27-year-old US left-hander, allowed one run on six hits with two walks and two strikeouts over five innings.
"They made me work a lot," Snell said. "They made it really tough. The bullpen held it down like always bit a lot more work to be done."
John Curtiss and Ryan Thompson each pitched an inning of shutout relief and Castillo closed down the Astros in tricky situations in the eighth and ninth innings, delivering Tampa Bay's 60th consecutive victory when leading after seven innings.
The Astros tagged Rays relief pitcher Aaron Loup to start the eighth. Houston's Michael Brantley was hit by a pitch, took second on a wild pitch and, after a one-out Carlos Correa walk, took third on a bases-loading single by Kyle Tucker.
That prompted Tampa Bay to bring in Dominican right-hander Castillo, whose first pitch -- a 97 mph sinker -- enticed Yuli Gurriel to hit into an inning-ending double play to second baseman Brandon Lowe, halting the Astros threat and keeping the Rays ahead.
"It hurts to lose these one-run games," Astros manager Dusty Baker said. "You don't have any chance if you don't get any opportunities. We had a heck of a lot of chances tonight and they just got away."
In the bottom of the eighth, Tampa Bay's Yandy Diaz opened with a single and Manuel Margot walked with two outs, but Kevin Kiermaier popped up to end the threat, the Rays falling to 1-for-8 at the plate with runners in scoring position.
"There's no margin for error and our guys take that approach every time they take the field," Rays manager Kevin Cash said.
In the ninth, Castillo surrendered a one-out single to Josh Reddick and George Springer advanced the runner into scoring position on a ground out only to have Castillo strike out Altuve to end the game.