Zobrist doubled in his first at-bat before crushing a grand slam off Twins right-hander Scott Baker in the second inning.
Baker, making his first start since March 6 due to elbow tendinitis, struggled against the Rays. He allowed seven runs on seven hits and a walk over 2 2/3 innings.
"Our starting pitcher put us in a hole today, and got his work in, but we chased the ball all over the field," Twins manager Ron Gardenhire said. "It was like eight hours in the first three innings out there. That's not good. He has to pick up the pace and get going. He says his arm is fine and everything is good."
Johnson hit his two-run homer off Baker in the third, after Jeff Keppinger singled home Evan Longoria earlier in the inning.
Rays right-hander Jeff Niemann allowed two runs on six hits and three walks over five innings. He struck out five.
Ryan Doumit hit a solo shot off Niemann in the second inning to mark his first homer of the spring. The Twins added a run in the fifth, when Joe Mauer singled and Justin Morneau doubled before Doumit scored Mauer with a sacrifice fly to center field.
Cody Rogers hit a two-run homer in the eighth inning to extend the Rays' lead to 9-2. In the bottom of the frame, Luke Hughes' sac fly got a run back for the Twins.
Up next: The Twins trek to Sarasota to face the Orioles on Tuesday at 12:05 p.m. CT, with right-hander Anthony Swarzak scheduled to make his first start of the spring. Swarzak has only been used in relief so far, posting a 6.35 ERA in 5 2/3 innings. Swarzak, who is out of Minor League options, is being stretched out in case he's used as a long reliever to start the season. Matt Capps, Glen Perkins and Matt Maloney are also scheduled to pitch for the Twins.