Davis drove in two runs during a five-run third inning that ruined Price's outing and blasted the decisive three-run home run off Kimbrel in the ninth, helping the Baltimore Orioles stay perfect with a 9-7 win over the Boston Red Sox.
"We knew we were going to have our work cut out for us coming in here," said Davis, who was 2-for-5. "First road trip, opening day for those guys. They were going to be jacked up.
"When you've got a guy like David Price on the mound, you know you've got to battle. Any win on the road is good, especially early in the season."
Davis unloaded on an 0-1 knuckle-curve from Kimbrel and deposited it into the center-field bleachers after Caleb Joseph and Manny Machado walked to snap the 6-6 tie.
Ortiz grounded into a double play and Hanley Ramirez struck out to end the game.
"Any time you're playing an AL East opponent, you know that no lead is safe," Davis said. "Especially with the lineup that those guys have."
The Orioles' first baseman also drew praise from his skipper for his work in the field.
"I thought one of the keys to the game was the game that Chris Davis had at first base defensively," Orioles manager Buck Showalter said. "That's as good as you'll see a first baseman play nine innings."
Baltimore's defensive play behind him wasn't as strong, as the Orioles committed a pair of fielding errors in the outfield.
"The outfield's so hard here as it is, but when you play these 2 o'clock games, the sun and the shadows, it's tough to see," Showalter said.
Mark Trumbo also clubbed a three-run homer and Jonathan Schoop added an RBI for Baltimore (6-0), which remained the only undefeated team in the majors.
"Our guys are very reality-based, understand it doesn't mean much of anything," Showalter said. "It's just six games that we don't have to win now."
Yovani Gallardo gave up five runs on seven hits and a walk while striking out three in a five-inning no-decision for the Orioles.
Brad Brach (2-0) earned the victory and Zach Britton picked up his third save.
Betts and Jackie Bradley Jr. each drove in two runs, while Ortiz, Ramirez and Bogaerts had one RBI apiece for Boston (3-3).
Price was up and down in his Fenway debut with the Red Sox, striking out eight but allowing five runs -- all in the third -- on five hits and two walks over five innings.
"I enjoyed having the fans behind me," Price said. "Even after I gave up that five-spot, they were still behind me. It was huge. It didn't go the way I wanted it to, but that's part of it."
Kimbrel (0-1) took the loss after serving up the Davis homer and had two strikeouts.
"It happens. We are all human," Price said about Kimbrel. "He's one of the best closers in baseball and we are definitely happy he's on our team. If it's a save situation, he's going to be right back out there."
Baltimore broke a 5-all tie on Schoop's RBI double in the sixth, but Bradley beat out a throw at first to plate Brock Holt in the bottom of the frame to make it 6-6.
Bogaerts kicked off Boston's three-run first inning with an RBI single up the middle.
Ortiz, who was given a standing ovation before his first at-bat, showed his appreciation with an RBI single off the Green Monster in left, and Ramirez's ensuing sacrifice fly made it 3-0.
Davis avenged a first-inning strikeout -- Price's first with Boston at Fenway Park -- with a two-run single with the bases loaded and nobody out in the third.
Trumbo's three-run blast to center one at-bat later gave Baltimore the lead at 5-3.
"It was execution in that third inning," Price said. "That has been my Achilles heel. That one bad inning. That's all it takes in this game. One bad pitch. Today it was just that one inning."
Bradley's ground-rule double into the stands in left let Holt score in the fourth, and Betts brought him home on a fielder's choice as the Red Sox knotted the score at 5-5.
NOTES: Red Sox DH David Ortiz, who said he plans to retire following this season, was accompanied by Boston sports icons Bill Russell, Bobby Orr and Ty Law to throw out the ceremonial first pitches. Ortiz's daughter, Alex, sang the national anthem. ... Boston's John Farrell managed his first game at Fenway Park since being diagnosed with stage one non-Hodgkin's Burkitt lymphoma on Aug. 14, 2015. On Oct. 22, Farrell said his cancer was in remission. ... Orioles CF Adam Jones (rib) entered as a defensive replacement in the ninth inning after missing three straight games. Jones ran and played catch before the game. ... Baltimore is enjoying its best start since 1944, when they were the St. Louis Browns and opened 9-0.