Derek Jeter had three hits and collected three RBIs for the Yankees, who won for just the fourth time in 13 games. Phil Hughes (11-8) pitched six innings of one-run ball for the victory as the Yankees restored their lead in the American League East to 6 1/2 games over Baltimore.
"Every game means a lot," Cano said. "We're now in the second half of the season, when one game means two games. You don't have too much time, and you just want to keep the lead so that when September comes, you don't have to wait for somebody to lose."
The third-inning barrage was marked most by Cano's ninth career grand slam -- and second of the season -- a moonshot to right off Baltimore reliever Kevin Gregg. Jeter also ripped a two-run double in the inning, part of a three-hit performance from The Captain.
New York peppered starter Zach Britton (1-1) for seven runs in 2 2/3 innings. Curtis Granderson hit his team-leading 29th homer and Andruw Jones provided a first-inning sacrifice fly to produce a quick lead, while Jeter and Swisher ripped run-scoring singles off Britton in the second.
"They started out with 11 runs in the first three innings, and it was tough to come back," Baltimore's Adam Jones said. "We put up  hits, we just didn't get hits with men in scoring position, but we swung the bats fine. We got the series. We're happy about that. We aren't content, but we're happy about it."
Britton permitted three of the five batters he faced in the third inning to reach base, and Gregg was unable to close the inning, permitting two hits, a walk and Cano's grand slam -- the Yankees' Major League-leading eighth of the year -- before finally escaping.
"We have scored some runs in the midst of our struggles," Yankees manager Joe Girardi said. "It's not like we've been getting shut out. We've swung the bats pretty good. Even with some guys out, this offense is still capable of scoring some runs."
Hughes didn't leave the bullpen before the game feeling like he had his best stuff on a muggy, rainy afternoon, but he had the luxury of breezing through his six frames. He scattered nine hits while limiting the Orioles to one run, coming on Endy Chavez's second-inning RBI groundout.
"I didn't pitch well at all," Hughes said. "I didn't have good stuff -- didn't have good location. I just tried to bear down and get some outs when I really needed them. I was really fortunate to get the run support. I was feeling good after my last start, so it's kind of unfortunate to come out and have a start like this, but we got a lot of runs, so that's nice."
Hughes threw 107 pitches, walking two and striking out two as he won his second straight start. He did not permit a homer for the fifth time this season, and his teammates held a more positive view of the outing.
"Phil's one of those guys where, if you can get him a lead, he can utilize that fastball," Swisher said. "In a game like that, you're not necessarily looking to dot corners, you're looking for the middle of the plate. Today, with the way the conditions are, it was a great game for us, and it definitely makes that [Thursday] off-day a lot better."
Owning a huge advantage, the Yankees found a perfect time to send Joba Chamberlain back to a big league mound for the first time since last June 5 in the seventh inning.
The outing could have gone better for Chamberlain, who surrendered a home run on the second pitch he threw, a J.J. Hardy solo blast. He yielded two runs on four hits and a walk over 1 2/3 innings of work.
"Obviously that wasn't the result I wanted, but I have to look at the picture as a whole and to see what I've come back from in the past 14 months," said Chamberlain, who returned after Tommy John surgery as well as an open dislocation of his right ankle.
Jayson Nix also had three hits and an RBI for the Yankees, and was one of four Yankees to score two runs; Casey McGehee, acquired on Tuesday from the Pirates, also scored twice and added an eighth-inning sacrifice fly in his pinstriped debut.
Ichiro Suzuki played left field for his first time in a regular-season game, extending his hitting streak to nine games with a fourth-inning single. Ichiro has notched one hit in each game he has played as a Yankee.
"We feel good right now with where we are," Swisher said. "We're still a little banged up. To be putting up the numbers that we are with the injuries we've dealt with, I think that says a lot about this team."