Swisher hit a tying home run in the eighth, then lined a single that sent Curtis Granderson sliding home for the victory in a match-up of the teams with the best records in baseball. The Yankees streamed from the dugout to celebrate, and Swisher wound up way out in right field, surrounded by jumping teammates.
The joyful ending was in sharp contrast to earlier events that honored Steinbrenner, who died Tuesday, two days after the passing of Sheppard.
A 15-minute pre-game tribute included a 2-minute moment of absolute silence. Not a single fan shouted out, and the only sounds were the flags flapping at half-staff and a passing subway train.
Granderson opened the bottom of the ninth with a single against Randy Choate (2-3). Ramiro Pena sacrificed, Brett Gardner walked and Derek Jeter struck out. Swisher lined a single and Granderson beat right fielder Gabe Kapler's throw.
Robinson Cano and Jorge Posada hit consecutive home runs off James Shields in the sixth to tie it at 3.
At Boston, Bengie Molina hit for the cycle with a grand slam and triple in his last two at-bats as Texas came up with another big inning in a victory over Boston.
Molina's slam came in a five-run fifth inning and gave Texas a 7-3 lead. One night earlier, he capped a six-run first inning with a two-run homer and the Rangers went on to a 7-2 win.
The slow-footed catcher, obtained from San Francisco on July 1 to shore up a team weakness, lumbered into third base with a leadoff drive over the head of center fielder Eric Patterson and smiled at teammates in the dugout. A few pitches later, he left the game with tightness in his right quadriceps and was replaced by pinch-runner Joaquin Arias.
Arias scored the final run of the game on a sacrifice fly by Michael Young.
Molina is the fifth player in Rangers history to hit for the cycle and the first since Kinsler on April 15, 2009.
The last visitor to hit for the cycle at Fenway Park was Cleveland's Andre Thornton on April 2, 1978.
The AL West-leading Rangers lost their last four games before the All-Star break but returned to take the first two of a four-game series against the Red Sox, who have lost seven of nine as injuries severely weakened their lineup.
At Minneapolis, Minnesota snapped Chicago's nine-game winning streak with the help of a strong outing from Francisco Liriano.
Liriano (7-7) left with two outs in the eighth inning. The Twins had lost Liriano's last five starts.
Liriano also got some help from Chicago's defense. The White Sox made four errors, and Gavin Floyd (5-8) gave up four runs in the fourth inning.
Joe Mauer's two-run single in the ninth gave Jon Rauch a five-run pad he needed in the bottom of the inning, when he walked three and surrendered two hits before being pulled for Jesse Crain with one out and the bases loaded. Crain got his first save by getting the last two outs, striking out Paul Konerko on three pitches.
At Cleveland, Andy Marte and Austin Kearns hit two-run homers as Cleveland trounced Detroit.
Marte broke at 2-2 tie in the fourth inning against Max Scherzer (6-7). Kearns' eighth homer of the season capped a four-run sixth against the Tigers' bullpen that put Cleveland ahead 8-2.
Jake Westbrook (6-5) gave up two runs over 5 2-3 innings to win his first home start against Detroit since Sept. 18, 2007. The right-hander missed most of 2008 and all of 2009 following reconstructive elbow surgery.
Miguel Cabrera extended his hitting streak to 20 games with a run-scoring single that put the Tigers ahead 1-0 in the first. Austin Jackson singled to open the game, stole second, and scored on Cabrera's two-out liner off the left-field wall.
At Baltimore, Ricky Romero allowed five hits in seven innings and Aaron Hill homered as Toronto ended Baltimore's four-game winning streak.
Fred Lewis had two hits and scored twice for the Blue Jays, who improved to 7-0 against the Orioles this season, including 4-0 in Baltimore. Toronto was 1-8 at Camden Yards last year.
Romero (7-6) gave up two unearned runs, walked one and struck out five. He lost his previous three starts, yielding a combined 13 earned runs over five innings in the last two.
Hill had three hits, including Toronto's major league-leading 137th home run. The Blue Jays have homered in nine straight games, collecting 22 over that span.
At Kansas City, Missouri, Gio Gonzalez outdueled Zack Greinke as Oakland took advantage of the umpires' reversal of a call in the first inning.
Kevin Kouzmanoff had a two-run single for the Athletics while Gonzalez (8-6) allowed seven hits and one walk in seven innings. Relievers Craig Breslow and Michael Wuertz each pitched a shutout inning.
Leading off the game, Oakland's Coco Crisp was awarded a double after first base umpire Larry Vanover first ruled a foul ball on Crisp's liner down the right field line. On replays, it did appear to come down an inch or so fair. A's manager Bob Geren came out to argue, the umpires huddled, and the call was reversed.
Jack Cust drove in Crisp with a sacrifice fly.
