Heybridge came close to opening the scoring in their first ever visit to the QE Stadium when Broadhurst placed a through ball into the path of Jack West but the keeper Imber saved.
Enfield Town took a 19th minute lead when a poor goal kick from Jason Willis was fed back by Enfield into the path of Adam Wallace who knocked the ball past the advancing keeper before going flying in the box with the ball flying wide. The Enfield man made the was awarded a penalty, although Willis had got in the way it was unclear how much contact there was if any and the keeper stayed on the field with a yellow card. Liam Hope despatched the penalty for 1-0.
Swifts were out of the blocks with enthusiasm in the second half and showed plenty of creation with early shots by Callander saved and then a head height drive from Russell Pond.
Leon Osei was denied by the post for Enfield just before the hour with a run and shot and from the rebound Dean Pennant missed a sitter.
Luke Callander went on a search for an equaliser and saw another shot saved on 71 minutes after goodwork from Dobinson, then seven minutes later Ofori shoots over after Callander had won the ball back in midfield and run up towards the area.
The equaliser did come on 82 minutes when a lobbed free kick from halfway in the hosts half was bundled home at the far post by Luke Callander after a posse of players had got touches but before the crowd had chance to take breath Enfield raced to the other end and regained the lead with a side footer from 12 yards by Sub Michael Ewang along the ground after a cross from the left.
Michael Ewang became provider for Enfield's third on 90 minutes as Liam Hope netted his second of the game.
Heybridge almost made for an interesting stoppage time but the crossbar denied a 30 yard free kick from Luke Callander which had Enfield keeper Imber motionless but the woodwork meant Enfield weren't to have any late nervy moments.
It was still a positive performance by Heybridge who will need to recreate their forward thinking moves which will surely pay dividends in more than most games.