Burnett (1-0) allowed two runs in 5 1-3 innings in his New York debut, Swisher went 3-for-5 with a homer and a career-high tying five RBIs, and Teixeira had two hits and a home run.
Robinson Cano homered and scored four runs for the Yankees, who avoided their first 0-3 start since 1998.
After Teixeira and Swisher hit their first home runs in the third inning to give New York the lead for good, the Yankees used a four-run sixth to go up 7-2. Cano added a two-run shot in the seventh, and Swisher doubled in two runs in the ninth.
Luke Scott homered for the Orioles, denied their first three-game sweep of the Yankees at home since 2005.
Signed to a $82.5 million, five-year contract during the offseason, Burnett ended a series of ineffective performances by New York starters. With CC Sabathia and Chien-Ming Wang on the mound, the Yankees allowed seven or more runs in their first two games for just the third time, according to the Elias Sports Bureau, following 1973 and 1985.
Sabathia and Wang failed to strike out any batters, the first time Yankees starters did that in consecutive games since Sept. 1-2, 2005, according to Elias.
Burnett, the 2008 AL strikeout leader, fanned six, allowed seven hits and walked one. He improved to 8-1 against the Orioles, including 4-0 in five starts at Camden Yards Teixeira hit a fly ball in his first at-bat against Alfredo Simon (0-1) to drop to 1-for-10, then homered leading off the third and singled and scored in the sixth. For the third straight game he was booed by Orioles fans for signing as a free agent with New York instead of accepting an offer from his hometown of Baltimore.
Swisher, obtained in an offseason trade with the Chicago White Sox, hit a two-run drive in the third and singled in a run in the seventh.
Baltimore went up 1-0 in the third on a RBI single by Brian Roberts, then loaded the bases with two outs. Burnett then struck out cleanup hitter Aubrey Huff, and pumped his fist in delight at ending the 28-pitch inning in positive fashion.
Scott's first homer got Baltimore to 3-2 in the bottom half, but New York used four hits and a costly error to pad the cushion in the sixth. After Teixeira chased Simon with a leadoff single, Cano singled off Brian Bass and Swisher singled in a run. Cody Ransom walked to load the bases, and Brett Gardner hit a comebacker to Bass, who threw wildly to the plate, allowing two runs to score.
Simon was charged with four runs in five-plus innings. He allowed only seven runs in 18 2-3 innings during spring training.
Notes: New York's Ramiro Pena got his first major-league hit in the ninth inning. ... Orioles reliever Chris Ray gave up two runs in the ninth, lifting his ERA to 27.00 compared to 0.00 in spring training. ... Mariano Rivera worked a perfect ninth for the Yankees.
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