"It was a good day of pounding the right part of the strike zone," Dempster said. "I got ahead of most of the hitters and just tried to execute pitches."
The Red Sox got a quick run in the bottom of the second when Mike Napoli hit a mammoth home run over the 420-sign in right-center. It was Napoli's first homer of the Grapefruit League, and it came in his second game.
That homer by Napoli was the only hit allowed by Yankees Minor Leaguer Adam Warren, who struck out two in three innings of work.
For Boston, Napoli's homer held up until the top of the sixth, when the Yankees capitalized on two errors by third baseman Drew Sutton.
Closer Joel Hanrahan was on the mound for Boston in that sixth, and he gave up an RBI single to Eduardo Nunez. Hanrahan went two-thirds of an inning and walked two. Of the three runs he allowed, just one of them was earned.
Lefty Andrew Miller turned in a dominant inning for the Red Sox, striking out all three batters he faced. Righty Koji Uehara was also solid in his scoreless inning.
Though Dempster looks forward to joining the Red Sox-Yankees rivalry, he didn't get much of a taste of it in this one, as most of New York's star players are either at the World Baseball Classic or injured.
"I've always enjoyed it from afar, being a baseball fan," Dempster said. "I'm sure we'll have lots of good games against each other. I think for us as a team, we just want to play hard and play right, no matter who we're playing. I know that rivalry is different than any rivalry in baseball, and pretty much one of the best ones in all of sports."
Up next: Felix Doubront, who was slowed by some weakness in his left shoulder, will make his Grapefruit League debut in Monday's 1:35 p.m. ET home game against the Rays. Daniel Bard, who didn't pitch for the last week so he could work on some mechanics, will see action in a simulated game.
Ian Browne is a reporter for MLB.com. Read his blog, Brownie Points, and follow him on Twitter @IanMBrowne. This story was not subject to the approval of Major League Baseball or its clubs.