Corey Dickerson hit two of Tampa Bay's season-high five homers, including a 449-foot blast that landed in the trees beyond the center field wall in the fifth.
"It was one of my top ones, to be able to hit it clean like that and to know it's gone," he said. "The ones to dead center always feel good. I thought it was further than 450, though."
"I know we got a bunch of solo shots in there, and then Beck came up with a big two-run blast," Rays manager Kevin Cash said. "The offense definitely carried over into this game and got us a nice cushion there for Odo to work with."
Indians center fielder Bradley Zimmer was 0 for 3 and struck out three times in his major league debut. Cleveland called up its No. 1 pick in 2014 from Triple-A Columbus earlier Tuesday.
The Rays won despite striking out 16 times.
Rasmus homered with two outs in the second while Norris and Dickerson went back-to-back to start the third. Beckham's sixth-inning home run pushed the lead to 6-1.
After using five relievers in Monday's win, the Indians needed Salazar to pitch deep into the game, but he was pulled following a leadoff double in the sixth. The right-hander was charged with five runs and struck out nine in five-plus innings.
"I'm going to give you a couple of adjectives," Indians manager Terry Francona said when asked about Salazar's performance. "One's vexing. He has good enough stuff to punch out nine. Gives up six hits, four of them are home runs. When he made a mistake he really paid for it."
Salazar was coming off a shaky outing against Toronto, in which he couldn't hold two early leads and allowed five runs in 2 2/3 innings. He hasn't won since April 29.
"I've been struggling," Salazar said. "Today, I was able to bring my ball down a little bit, but things just aren't working the way I'm expecting."
With Zimmer playing his first game, Francona recalled his major league debut with the Montreal Expos against the Houston Astros on Aug. 19, 1981. He was scheduled to be in the lineup against Nolan Ryan, but a strike by air traffic controllers delayed his flight from Denver, where he had been playing in Triple-A, to Houston.
Ryan was pitching a no-hitter when Francona arrived at the Astrodome in the fourth inning. He pinch hit in the eighth and grounded out against Dave Smith.
Rays: 2B Brad Miller (strained left abdominal muscle) was scratched about two hours before the game.
Indians: RHP Carlos Carrasco (strained left pectoral muscle) will resume his between starts routine in a couple of days. He's scheduled to start again on May 23 against Cincinnati.
Rays: RHP Alex Cobb is 2-2 in four regular-season starts against Cleveland. He pitched 6 2/3 scoreless innings and got the win in 2013 AL wild card game at Progressive Field