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Maidstone are off to Cheltenham in the FA Cup first round after an extra-time victory at Bostik League Enfield.
Hines, who had one penalty chalked off and another saved in the first game, gave United the lead just after the break.
United should have been through when they were awarded their third penalty of the tie midway through the second half but this time Joe Pigott’s strike was saved by Joe Wright.
It proved costly as Enfield scored an unlikely equaliser through Aaron Green to force extra-time.
Finney restored United’s lead from Hines’ corner and substitute Jai Reason added United’s third.
Enfield, much like in the 2-2 draw at the Gallagher, worked hard to close Maidstone down and stop them playing.
If they were tired from the big effort on Saturday, it didn’t show as they set about making life difficult for a Stones side who always had that extra quality in possession.
The Towners had a bit about them going forward, too, Finney alert to cut out a couple of dangerous crosses while a big early chance went begging when Micky Parcell found the unmarked Ryan Blackman, who blazed over.
There wasn’t much space for Maidstone to play with in attack but, driven on by Stuart Lewis, they did carve out some decent chances.
The best fell to Delano Sam-Yorke whose shot was blocked on the line from a few yards out following a corner.
Sam-Yorke also bent an effort wide after swapping passes with Pigott on the edge of the box.
Crosses from Sam-Yorke and Pigott just evaded Tom Wraight, who’d got himself in the six-yard box, while Joe Anderson went for placement rather than power from Hines’ cut-back.
United led for the first time over the two ties in the 47th minute, Pigott finding Sam-Yorke who squared for Hines two yards out.
United were now well on top, Hines’ clever pass playing in Wraight but the winger missed the target when all alone.
Maidstone had the chance to make the tie safe when Sam-Yorke was brought down in the box in the 65th minute.
Pigott was back on spot-kick duty after Hines’ weekend woe but Wright did brilliantly to push his effort on to the bar.
Enfield equalised out of nothing in the 73rd minute. They broke quickly after Hines’ poor free-kick hit the wall and seconds later the ball was with Greene who fired low into the far corner.
The goal lifted an Enfield side who had started to look tired, Blackman and substitute Montell Moore firing over after balls dropped in the box, then big Simon Thomas held off his man, turned smartly and forced a good save from Lee Worgan.
Stones were back in front in the 102nd minute, Hines’ corner flicked on and Finney stabbing home at the far post.
And the tie was in the bag in the 108th minute when substitute Jai Reason broke into the box and finished smartly in the corner.
Sam-Yorke thumped an effort against the bar as United threatened a fourth, with Reason’s follow-up blocked.
Maidstone: Worgan, Hare, Anderson, Wynter, Finney, Muldoon (Prestedge 55mins), Lewis, Hines (Reason 106mins), Sam-Yorke, Pigott, Wraight (Loza 79mins). Subs not used: Paxman, Twumasi, ter Horst, Richards.
Attendance: 820.