Brandon Phillips had given the Reds a 3-2 lead with a two-run homer off Dillon Gee in the eighth, two innings after Curtis Granderson connected for a two-run drive off Johnny Cueto, his first homer with the Mets.
Anthony Recker bunted and first baseman Joey Votto threw to second base for the force play. Second base umpire James Hoye called Lagares out and Mets manager Terry Collins raced out of the dugout to ask for a challenge. After a review of 2 minutes, 15 seconds, the call was overturned by crew chief John Hirschbeck.
Davis, having lost his starting first base job to Lucas Duda a day earlier, was called on to hit in the pitcher's spot and lined a 0-1 pitch to right field, setting off a gleeful celebration at Citi Field.
The Mets won their second straight after being swept by Washington in three games to open the season. Cincinnati dropped to 1-4 for the first time since 2003, when it finished 69-93.