CINCINNATI >> First-year manager Mickey Callaway knew something was badly wrong when Wilmer Flores went to the plate for the first time. At that point, it was too late to change the Mets’ mistake.

New York wasted a first-inning rally by batting out of turn, and Adam Duvall led off the 10th inning with a homer that sent the Cincinnati Reds to a 2-1 victory today, concluding a series that will be remembered for an unexpected trade and an embarrassing mistake.

This one simply wasn’t in the lineup cards for New York.

“It is frustrating,” said Callaway, who blamed himself for an administrative mistake. “It probably cost us a game. We had a chance to score in the first and we didn’t.”

The Reds took advantage. Duvall connected for the first game-ending homer of his career off A.J. Ramos (1-2), sending New York to its eighth loss in nine games. The Mets have scored two runs or less in seven of them, including three shutouts.

The Reds took two of three for only their second series win of the season. They’re 10-27, their worst start since the Great Depression. Raisel Iglesias (1-0) retired six straight batters, fanning three.

It started with a miscommunication. The Mets handed Callaway’s lineup card to the umpires, which had Asdrubal Cabrera batting second and Wilmer Flores third. However, the lineup sent out by computer pregame and posted on the dugout wall incorrectly had the two hitters switched in the order. Callaway did not identify who sent out the incorrect lineup.

Callaway knew there was nothing he could do when Flores batted out of turn and struck out for the second out. Cabrera then doubled to left, and the Reds pointed out the discrepancy. The double was nullified and an out was called , credited to Jay Bruce, the next batter.

“I felt bad,” Reds interim manager Jim Riggleman said . “It’s so easy to have that happen.”

Crew chief Jerry Meals had never been involved with a batting-out-of-turn in the majors.

“I’ve seen it in the minor leagues,” Meals said. “It’s been a long time, 32 years.”

It was the first time a major league team was caught batting out of turn since the Brewers did it in 2016. The previous time the Mets batted out of order was 1977.

Following their correct order the next time through, the Mets got a run in the third when Brandon Nimmo tripled off Sal Romano and scored on Cabrera’s groundout.

Zack Wheeler had retired 12 straight before the Reds rallied in the sixth to tie it on Joey Votto’s single. Wheeler had to wait until the umpires sorted out the batting-out-of-order situation to throw his first pitch.

“I don’t know,” Wheeler said. “Some weird baseball. Unfortunately it happened, but it does happen every once in a while.”

Catcher Devin Mesoraco started for the Mets and went 0 for 4, a day after the Reds dealt him to New York for pitcher Matt Harvey.