Adrian Beltre added a solo homer for Boston, swept in a three-game series by the Chicago White Sox last weekend.
The Rays lost their third straight and remained 2 1/2 games behind the first-place New York Yankees, who lost to Baltimore 4-3 earlier in the day.
Boston moved within 6 1/2 games of Tampa Bay in the wild-card standings.
Lester (16-8) gave up two runs and four hits while striking out 10 and walking three for his fifth win in six decisions.
Boston took control by scoring three runs in each of the first two innings, and broke it open when Kalish hit his slam in a five-run fourth.
Ortiz and Beltre hit consecutive homers three pitches apart in the first to put the Red Sox ahead 3-1.
In the second, Victor Martinez had a two-run single to chase Jeff Niemann (10-6). Ortiz greeted Andy Sonnanstine with a double off the Green Monster, making it 6-1.
Niemann got just five outs, getting tagged for six runs, four hits and three walks. He's been roughed up in all three starts since coming off the disabled list from a strained shoulder on Aug. 25, allowing 23 runs in 16 2/3 innings.
Boston turned it into a rout by adding five off Sonnanstine in the fourth. After loading the bases on singles by Daniel Nava and J.D. Drew and a two-out intentional walk to Beltre, Sonnanstine walked Jed Lowrie to force home a run before Kalish belted his slam into the Rays' bullpen in right, making it 11-2. It was his second grand slam of the season, tying a club record for rookies held by six others.
The Rays scored three runs in the seventh to cut it to 11-5 against reliever Robert Coello, making his major league debut. Scott Atchison came in and got Reid Brignac and Rocco Baldelli to pop out to end the inning with the bases loaded.
Coello gave up three hits and two walks while retiring just one batter.
Carl Crawford's RBI single gave Tampa Bay a 1-0 lead in the first.
NOTES
Bob Zupcic was the previous Red Sox rookie with two grand slams in a season, doing it in 1992. … Red Sox manager Terry Francona didn't come right out and say it, but he indicated that All-Star RHP Clay Buchholz will pitch Wednesday only if Boston wins the first two games of the three-game series. "I think we reserve the right to see how we do," Francona said. "I think it's obvious what I'm saying there. The plan now is to pitch Buch." … Carlos Pena is 0-for-19 since homering in consecutive games Aug. 29 and 30. … The Rays recalled Sonnanstine before the game from Class-A Hudson Valley. He was sent down to make room for OF Brad Hawpe on Aug. 31. … Boston SS Marco Scutaro is scheduled to have an MRI on his right shoulder Tuesday. It's been bothering him for about a month. … Boston recalled C Jason Varitek, who was on the DL with a broken right foot since July 2, after a two-game rehab assignment at Triple-A Pawtucket. … The Red Sox honored former RHP Luis Tiant for his upcoming 70th birthday. They gave him a watch and he threw out the ceremonial first pitch. Former players Johnny Pesky and Dwight Evans were among those on hand.