Tomlin (2-0), who pitched just once in the big leagues last season after right elbow surgery in 2012, allowed two runs and six hits over six innings in his second outing this year.
Morgan had an RBI infield single that rolled to a stop near the line between home and third before Bourn hit a two-run double as the Indians took a 3-1 lead in the second. Morgan also hit his first major league homer since July 30, 2012, an eighth-inning solo shot that made it 6-2.
A three-run eighth pulled the Rays to 6-5.
Ben Zobrist scored the first run in the inning when reliever Marc Rzepczynski made an errant throw to second on what could have been an inning-ending double play. Cody Allen entered and, after allowing Wil Myers' RBI single, hit Desmond Jennings with a pitch to load the bases. Yunel Escobar hit a sacrifice fly.
Yan Gomes hit a leadoff homer that chased Archer during a two-run sixth that gave Cleveland a 5-2 advantage. Morgan reached on a bunt single off Brad Boxberger that involved a call overturned on replay, advanced on a balk and wild pitch, and scored the inning's second run on Mike Aviles' sacrifice fly.
Joyce hit a first-inning solo homer off Tomlin. Joyce, who is 6 for 12 with two homers against the right-hander, had an RBI single in the fourth.
NOTES: Cleveland manager Terry Francona tweaked his rotation for a three-game series at Toronto. RHP Danny Salazar (1-3) had his start pushed back to Thursday from Tuesday so that RHP Justin Masterson (2-1) and RHP Corey Kluber (3-3) can pitch Tuesday and Wednesday on normal rest. ... Tampa Bay RHP Jeremy Hellickson (arthroscopic right elbow surgery) hopes to resume throwing curveballs during bullpen sessions in a week. ... Indians 1B Nick Swisher was given the option by Francona to have a day off and took it. ... Rays LHP Cesar Ramos (1-1) and Mariners RHP Felix Hernandez (3-1) are Monday night's scheduled starters in Seattle. ... With the Rays' road trip theme being ''Woodstock,'' manager Joe Maddon looked the part from the 1969 concert, wearing a long-hair wig, sun glasses and tie-dye shirt.