NEW YORK — Tainted or not, Alex Rodriguez had plenty of reasons to celebrate homer No. 661.

A-Rod passed Willie Mays for fourth on the career home run list Thursday night, helping the New York Yankees beat the Baltimore Orioles 4-3.

“All of this is pretty crazy,” Rodriguez said. “A year ago today I never thought I would ever get a curtain call or be hitting in the middle of the lineup for the Yankees, helping our team win.”

With a loud crack of the bat, the Yankees star drilled a line drive off Chris Tillman into the ramp area off limits to fans just to the left of Monument Park in center field in the third inning. The milestone hit gave New York a 3-2 lead.

A-Rod sent a soaring shot in the first that Delmon Young caught by reaching over the top of the wall in right field for a sacrifice fly.

Next up for Rodriguez: Babe Ruth at 714 home runs.

The Yankees, who said they will not pay a $6 million bonus after Rodriguez matched the Say Hey Kid on Friday in Boston, put a message on the main videoboard saying, “661 Home Runs. Alex Rodriguez just surpassed Willie Mays for sole possession of 4th place on baseball’s all-time home runs list.”

The slugger pumped his fist as he rounded first base, and fans gave Rodriguez, who didn’t play last year while serving a drug suspension, a standing ovation as he was greeted warmly in the dugout by his teammates.

“It’s a lot of fun to be part of,” said Brett Gardner, who pulled off Rodriguez’s helmet. “I know there’s a lot of questions surrounding everything. Still, 661 home runs is a lot.”

Mark Teixeira stayed out of the batter’s box as the cheers swelled. Rodriguez was encouraged by the Yankees to take a curtain call, and he said he looked to manager Joe Girardi for reassurance that it was OK. Then, ever so briefly, the three-time AL MVP jumped to the top step and thrust both hands above his head.

“When you’re productive you re-earn their respect and they pull for you,” Girardi said of the fans.

Rodriguez, who turns 40 in July, moved into a tie for fourth in the AL this season with seven home runs.

The Yankees and A-Rod have a marketing agreement that calls for $6 million each for up to five achievements, payable within 15 days of designation by the team. But New York has said the marketing possibilities were ruined after Rodriguez served his performance-enhancing drug penalty for his involvement in the Biogenesis scandal.

The accomplishments were contemplated to be homers 660, 714, 755 (Hank Aaron), 762 (Barry Bonds) and 763 as he moved up baseball’s list.

No payment likely would trigger a grievance on Rodriguez’s behalf by the players’ union. Without a settlement, the case would be heard by an arbitrator.

“That’s not where my mind is right now,” Rodriguez said. “I’m really just trying to enjoy this moment.”

Rodriguez was just 4 for 21 since his pinch-hit homer into the seats above the Green Monster at Fenway Park gave New York the lead in the eighth inning Friday. To get that milestone long ball, Red Sox security traded several pieces of memorabilia autographed by David Ortiz.

Like his 600th homer on Aug 4, 2010, which landed in Monument Park, the 14-time All-Star had no trouble retrieving this ball.

Rodriguez singled in the seventh, closing within 38 hits of joining another elite club: 3,000 hits.

Teixeira’s RBI double in the fifth put New York ahead for good and kept Nathan Eovaldi (3-0) undefeated in six starts with the Yankees.

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