The first baseman Jackson's ninth inning throwing error allowed Carlos Beltran to score the Mets' go-ahead run from second.
"The ball just got away from me," said Jackson, praised by Arizona manager Bob Melvin for his improved defensive play this year. "I made a bad throw."
Meanwhile, the right fielder Church chased down the ball and threw perfectly to third, where David Wright tagged Burke out.
"The biggest thing was just getting the grip of it and letting it go, basically chuck it," Church said. "David did a great job of deking the runner a little bit, just grabbed it and snapped tag."
Melvin defended Hale's decision.
"Sometimes there's judgment involved and it just didn't work out," Melvin said.
Burke obviously wished he would have stayed at second.
"It was fortunate and unfortunate all in one play," he said, "just a frustrating play for the team because it could have set us up to take the lead, and then you never know what happens from there."
The Mets won at Arizona for the 15th time in 17 tries and handed the Diamondbacks' just their second series loss of the season. The only other series Arizona lost was its season-opening three-game set at Cincinnati.
Wright had a solo home run for the Mets.
"Any time a team makes mistakes on you, you make sure you make them pay for it," New York manager Willie Randolph said. "It was kind of a weird game. A few crazy things happened out there, but we scored early and late and got the job done."
Arizona's Mark Reynolds broke out of a 1-for-21 slump with a pair of bloop singles, the second bringing home the tying run in the seventh.
Reynolds blooped a broken-bat single to left to bring home Jackson and tie it a 2-2.
Santana allowed a run and six hits, striking out eight and walking four. Haren allowed two runs and only three hits. He struck out seven and walked one.
Reyes' two-out single, the first hit off Haren of the game, brought home Luis Castillo to put the Mets up 1-0 in the third.
New York made it 2-0 when Wright, leading off the fourth, hit Haren's 2-0 pitch deep into the left field seats, his sixth homer and 26th RBI of the season.
Arizona broke through with a run in the fourth. Upton singled, then scored when Chris Snyder lined one off the top of the bullpen fence in left, just inches shy of a home run, for his first career triple.
"Only the second series we've lost and it came right down to the ninth inning," Haren said. "You know, they're a good ball club, the Mets. I'm sure they'll be there at the end. I'm sure we'll be there at the end. We played them tough. They played us tough. This game could have gone either way."
Santana has allowed two earned runs in 22 innings against the Diamondbacks, including a complete game shutout at Arizona in 2005. ... Diamondbacks LF Eric Byrnes went 0-for-5 with three strikeouts to fall to 2-for-24 on the homestand. ... Diamondbacks pitcher Micah Owings had a pinch-hit infield single in the ninth.