Nick Markakis greets Chris Davis, right, after the latter’s first career grand slam, a second-inning blast that keyed a seven-run inning that rallied Baltimore to an 11-5 victory in New York. The Yankees had taken a 5-0 lead in the first inning. (Jim McIsaac/Getty Images)

Given the way Chris Tillman’s outing began Tuesday at Yankee Stadium, it appeared he was in for a short night. The Baltimore Orioles’ right hander gave up five runs a 36-pitch first inning as New York sent nine batters to the plate.

But Tillman’s teammates quickly granted him a mulligan, pouncing on pinstripe pitching for a seven-run second inning — capped by a Chris Davis’s first career grand slam — as the Orioles beat the Yankees, 11-5, in front of an announced crowd of 42,821 in the Bronx.

The Orioles have won four of five at Yankee Stadium this season and with the win, they cut New Yorks’ lead in the AL East to 5 1 / 2 games.

Earlier in the day, after Baltimore allowed Tuesday’s non-waiver trade deadline to pass without making a move, Manager Buck Showalter expressed that the Orioles’ aren’t just in the AL wild-card hunt. They want to compete with the Yankees for the division crown.

The Orioles fell into a whole early, only to be saved by their suddenly surging offense. The Orioles’ 11 runs were the most they’ve scored in a game in six weeks, when they scored 12 against the Pirates on June 14. All nine starters reached base and seven of them scored.

And after his rocky first inning, Tillman retired 13 of the next 15 batters and just one baserunner reached second base.

Tillman entered with a 12.27 ERA in three starts in the Bronx, with the Yankees hitting .448 (26-for-58) against him. And when seven of the first eight batters he faced reached base, it looked like a rough night for Tillman.

But Baltimore had an immediate answer, scoring all seven of its second-inning runs with two outs off right-hander Ivan Nova, the most the Orioles have scored in an inning since plating seven against the Red Sox in Boston on May 5.

Trailing 5-3, Davis launched a 0-1 hanging curveball to left-center field into the Orioles bullpen for his 18th home run of the season, suddenly putting Baltimore up two runs from down by two runs with one swing. Davis has three homers and 10 RBI in his last six games.

Before Davis’s shot, the Orioles pecked away at the New York lead. With two on and two outs, Mark Reynolds lined a ground-rule double down the left-field line to score Adam Jones from second. Omar Quintanilla then laced a single up the middle to plate two more runs to make it 5-3.

Nick Markakis’s single and a walk to J.J. Hardy loaded the bases for Davis, whose grand slam silenced the shocked Yankee Stadium crowd.

Markakis added three hits — including two-run homer in the eighth, his 10th of the year — and Wilson Betemit also had three hits, including two doubles, and drove in two.

The Orioles were 6-for-14 with runners in scoring position after going 3-for-17 in those scenarios in their previous three games combined. Over the last six games, Baltimore is averaging 6.2 runs – that after scoring four or fewer runs in 13 of the previous 16 games.

Nova allowed a career-high nine runs and 10 hits over five innings. In two starts against the Orioles this season, Nova has a 12.19 ERA, having allowed 17 hits and 14 runs over 101 / 3 innings.

Despite his forgettable first inning, Tillman won his third straight start.