Peter Fairbanks came in for Anderson and walked the first batter he faced and then allowed a two-run single to Carlos Correa.
The Rays advanced to Game 1 of the World Series in Arlington, Texas, on Tuesday night against the victor of Sunday night's NLCS game between the Braves and Dodgers.
Now the two teams will send out a pair of starters who both have excelled this postseason with Houston sending out young southpaw Framber Valdez and the Rays deploying former Cy Young victor Blake Snell.
Hinch's replacement, 71-year-old Dusty Baker, helped restore some of the goodwill the Astros squandered. Traded to Tampa Bay in the offseason, he tested positive for COVID-19 before summer camp and didn't make his 2020 debut until August 30.
Tampa Bay catcher Mike Zunino added a solo shot.
Until he started showing his October power, Arozarena was best known for a gaffe he committed while with the St. Louis Cardinals last postseason.
Tampa Bay starter Charlie Morton, who played for the Astros from 2017-18, had a two-hit shutout with two outs in the sixth inning when Cash came out of the dugout.
Anderson retired Michael Brantley to finish the inning and worked a scoreless seventh.
"Cash made the right move again - shocker", kidded ace Tyler Glasnow, now lined up to start the World Series opener.
The team awaits the victor of Game 7 of the National League Championship Series between the Los Angeles Dodgers and Atlanta Braves.
Tampa Bay also should have the advantage late if Game 7 is close after Houston used closer Ryan Pressly for the third straight night Thursday. Carlos Correa hit a two-run single off Pete Fairbanks before the reliever struck out Alex Bregman on a 100 miles per hour fastball.
Fairbanks finished the game off, leaving a runner stranded in the ninth.
McCullers Jr. lasted only 3 2/3 innings. He would later contribute a sacrifice fly in the sixth inning to score Ji-Man Choi. After going an ordinary 5 for 21 with runners in scoring position during their three wins over Houston, the Rays have just 10 runs and 20 hits in their three losses while batting 1 for 14 with runners in scoring position. On Friday night, he snapped his bat over his knee in frustration after striking out on three pitches.