Cleveland - Davis' hit off Tony Sipp (0-1) gave the Blue Jays a 5-3 lead and they held on to win another extra-inning game between the teams, who set a major league record by playing the longest opening-day game in history -- a 16-inning marathon Thursday.
Kelly Johnson homered for the Blue Jays and Toronto starter Brandon Morrow allowed one hit in seven innings.
Casey Janssen (1-0) pitched two innings for the win.
Jason Kipnis hit a two-run homer and Asdrubal Cabrera a solo shot for Cleveland.
Indians starter Ubaldo Jimenez, pitching as he appeals a five-game suspension, took a no-hitter into the seventh.
Edwin Encarnacion opened the 12th with a double off Sipp and went to third on Brett Lawrie's soft single to right. Davis, who entered the game as a pinch-hitter in the ninth, then lined his double to center, giving the Blue Jays a 5-3 lead. Davis stole third and scored on Colby Rasmus' single.
The Blue Jays overcame a three-run deficit in the ninth on Thursday to tie it. They only trailed by two this time, but tied it in the seventh off Jimenez before taking a 3-2 lead in the ninth on Johnson's homer off Vinnie Pestano.
Toronto had 39 come-from-behind wins a year ago, when the Blue Jays went 81-81. They've already got two in two games and manager John Farrell likes the way his team is building off last year.
"It is a continuance," he said before the game. "There is a very solid chemistry starting to develop in the clubhouse."
For seven innings, the teams were locked in a pitcher's duel as Jimenez and Morrow matched each other strike for strike.
Jimenez, who plans to drop his appeal for intentionally throwing at former Colorado teammate Troy Tulowitzki in an exhibition game, allowed one hit and two runs in seven innings.
Morrow was just as good, giving up just one hit over seven.
The Indians were two outs away from losing in nine innings when Cabrera, rewarded earlier in the week with a new three-year contract, connected off closer Sergio Santos, driving a 1-1 pitch over the wall in right.
It was only Cleveland's second hit, and it came at the perfect time for the Indians, whose bats have been cold even since they arrived from spring training in Arizona, where they also struggled to score. The Indians have just 12 hits in 28 innings.
Kipnis' homer in the fifth ended Morrow's no-hit bid and a scoreless drought of 18 2-3 innings for Cleveland.
Jimenez was nearly perfect for six innings and unhittable for almost seven.
He retired the first 17 Blue Jays in order before walking No. 9 hitter Colby Rasmus with two outs in the sixth. However, Jimenez got out of the inning by getting Yunel Escobar on an easy fly to right.
But in the seventh, a pair of walks and a wild pitch came back to haunt Jimenez, who gave up Lawrie's two-run, two-out single that tied it 2-2.
Jimenez walked three, struck out three and gave the Indians their second quality start in a row after Justin Masterson dominated the Blue Jays for eight innings in the opener.
Morrow was almost as good as Jimenez for four innings and should have been out of the fifth after getting two outs.
Casey Kotchman hit a dribbler in front of the plate that catcher J.P. Arencibia hustled to get but threw wildly to right of first baseman Adam Lind for an error. Kipnis then went to a full count before lining his first homer just over the right-field wall to make it 2-0.
Jimenez was suspended five games for drilling Tulowitzki in the left elbow during his final outing at spring training. After hitting Tulowitzki, Jimenez dropped his glove and charged toward home plate as both benches and bullpens emptied. The incident was prompted by comments made by Jimenez, Tulowitzki and other Rockies players in newspaper articles. Jimenez was traded to Cleveland at the deadline last season in a blockbuster deal.
Notes: Blue Jays INF Omar Vizquel is wearing No. 17 this season to honor his first manager, former All-Star shortstop Chico Carrasquel, a fellow Venezuelan. Vizquel has worn No. 13 most of his career and No. 11 as a tribute to Hall of Famer Luis Aparicio. "The cycle is complete now," Vizquel said. ... Indians OF Shin-Soo Choo took exception to a head-high fastball in Thursday's opener from Jays reliever Luis Perez and started toward the mound before being retrained. Acta said Choo might be a little more sensitive to inside pitches after having his wrist broken by a pitch last season. "He's going to have to live with it," Acta said. "He understands that. It goes back to what happened last year. He also knows how important he is to us and it's why he stopped and handled things well."