The Mets played there first extra inning game at Shea Stadium on May 31st 1964 against the San Fransico Giants. This was the 2nd game of a double header at Shea that Sunday. The first game lasted 2 hours and 30 minutes. The Mets had lost the first game by a score of 5-3.
The Mets tied the score in the second game when they scored 3 runs in the bottm of the 6th. Amazingly no one would score until the top of 23rd when the Giants scored 2 runs and won the game 8-6. The game took 7 hours and 23 minutes to complete.
Couple of Notes of Interest in this game. The Mets had a triple play by then Shortstop Roy McMillian. A young Gaylord Perry pitch many of the extra innings for the Giants.
Picture here is from Giant game in 1964 . Thanks to the people over at nyc parks.org . Notice the giant Mets logo has not been placed yet on the scoreboard.