Wanderers, who are in the Thurlow Nunn Premier League, looked down and out at half-time as they trailed by three goals - two Rob Laney strikes and Carl McCluskey hitting the net for the visitors.
Craig Jennings pulled one back before Marc Weatherstone, who was also on target in the original tie which finished 2-2, restored Wingate & Finchley’s three-goal advantage.
The FA Cup
First Round Qualifying Replays
Wednesday 17 September 2014
Winning club receive £3,000
By Nick Callow
Refusing to roll over, the hosts hit back almost instantly as Ross Myhill made it 4-2 and Jack Severy set up a tense finish when he notched a third for Ipswich with half an hour to go.
But the miracle comeback was not to be and it is Wingate & Finchley who can now prepare for a tie against Hitchin Town after a stunning match at The Doucy.
Meanwhile, down on the south coast, just over 400 spectators were made to wait until the 85th minute for a goal when Ollie Rowe's effort for Lewes settled their tie with Bognor Regis Town.
And Micky Rae, who hit a hat-trick in the last round, was once again the goalscoring hero for Shildon AFC as he hit the winner in a 2-1 victory at Whitby Town.
Click here for The Second Round Qualifying draw in full.