Kelly lasted just 4 2-3 innings and gave up five runs for the third straight start as the Yankees completed a three-game series sweep over the Red Sox with an 8-5 win on Sunday night.
A month into the season the Red Sox are still searching for a stopper.
"For us to right our course, we still need the pitching from that rotation," Boston manager John Farrell said.
Kelly (1-1), who joked during the winter that he was going to win the Cy Young Award, had won his first three career starts against the Yankees before giving up two homers as Boston fell into an 8-0 hole.
The Red Sox lost for the eighth time in 11 games to fall below .500 for the first time this season. Mike Napoli hit a three-run homer in the sixth, the only inning Boston scored.
"There's nobody giving up in this clubhouse — not us," Kelly said. "Those guys are a good team. The hitters over there, they have the track record. That's a good lineup."
Jacoby Ellsbury had four hits, and Adam Warren took a one-hit shutout into the sixth inning for the Yankees, who swept the Red Sox in a series of three games or more for the first time since a five-game sweep in 2006.
Warren (2-1) allowed four runs in 5 2-3 innings. Andrew Miller struck out two and walked two before third baseman Chase Headley struggled to field Dustin Pedroia's grounder, loading the bases. Miller got David Ortiz on a line drive to center for his 10th save.
New York opened its 8-0 lead and then held on after Boston scored five runs and brought the tying run to the plate in the sixth.
Teixeira's two-run homer gave New York the lead in the first. The Yankees made it 5-0 against Kelly in the third on back-to-back doubles by Brian McCann and Carlos Beltran. And they took an 8-0 lead in the sixth on Gardner's three-run homer.
Warren, who didn't allow his first hit until Mookie Betts doubled to lead off the fourth, still had a one-hit shutout with two out in the sixth. But he gave up a single to Pedroia and an RBI double to Ortiz before hitting Hanley Ramirez with a pitch. Pablo Sandoval followed with an RBI single to left.
Esmil Rogers relieved Warren and Napoli hit his second pitch into the first row of the Monster Seats. Daniel Nava walked and Xander Bogaerts singled before Blake Swihart, who made his major-league debut on Saturday, struck out to end the inning.
Ramirez threw his bat aside and motioned angrily after being hit by the pitch. But it wasn't until Ellsbury was plunked in the eighth that the umpires warned both benches.
Red Sox RHP Dalier Hinojosa struck out Alex Rodriguez, the first batter he faced, and pitched 1 2-3 innings of hitless relief in his major league debut. He walked three and struck out two.
Red Sox: Clay Buchholz (1-3) will pitch the opener of a three-game series at home against Tampa Bay and try to earn his first win since pitching seven shutout innings on opening day. Jake Odorizzi (2-2) will pitch for the Rays.
Yankees: INF Brendan Ryan re-injured his hamstring while running in Florida during a rehab assignment. He is expected to rest for at least a week.
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