Joe Pigott's 95th minute equaliser made sure the Stones were at least in the hat for the first round draw with the winners this evening heading to League Two club Cheltenham Town.
Simon Thomas put the visitors in front in the 20th minute after Zavon Hines' penalty was disallowed for pausing in his run-up.
Hines then had a penalty saved before half-time but Alex Winter equalised moments later.
Enfield then had the opportunity to retake the lead in the 57th minute from the spot, and Darren Purse duly obliged. Then came heartbreak for Enfield as they were denied a place in the first round when Pigott tapped in with seconds remaining.
"It's good to still be in the hat," said Maidstone manager Jay Saunders. "We had a lot of possession and chances. Credit to Enfield, they've come here and worked their socks off. They were setup well and defended excellently.
"On another day we go through but it's the cup and it doesn't always work like that. We hit the post twice and played some good stuff. It's hard to break teams down when they come here and sit everyone behind the ball.
"Tuesday will be tough, they haven't lost at home. If we do the right things and take our chances, hopefully we can go through."
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