The 4-3 win helped Town cut the gap to second-placed Anstey to three points after Nomads were held to a 2-2 draw.
With the top two due to meet this weekend, Melton will make up further ground if they beat Burton Park on Saturday.
Town arrived at Bugbrooke just in time to see the home manager receive the UCL Division One December manager of the month award, underlining how hard this match was going to be.
And, for once, the visitors started slowly as Bugbrooke pounced after just five minutes.
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Goalkeeper Rob Peet collected a high free-kick, but the ball was hooked away to Moore whose cross back into the six-yard box found the tall centre-back Budd to touch the ball over the line.
Melton immediately responded when Nathan Arnold whipped a free-kick a foot wide, with Michael Reeve inches from a finish, while Lever was denied a clean run on goal by Boateng’s last-ditch tackle.
The Melton pressure eventually paid off when Hibbitt’s powerful header, from Arnold’s corner, was cleared off the line by Billey’s hand.
He was shown a red card, while Parsons put the penalty under Farrell to level.
Bugbrooke’s 10 men held their own for the rest of the half, with the only real chance coming when Arnold’s free-kick from the edge of the box looked bound for the top corner, but Boateng did well to head clear.
And from the resulting corner, Wilson headed just wide.
Bugbrooke adapted well at the start of the second half, while Melton over-played in midfield, and it came as no surprise when the hosts retook the lead on the hour after a series of corners.
Peet had already made two great saves to deny Daniel Porter and Hall, when Porter smashed a shot against the crossbar from the second corner.
But from the next, an unmarked Hall turned and smashed the ball into the net.
The game looked like to be slipping away from Melton, but with 17 minutes left, Westwood’s high cross was punched away by Farrell to Reek who took one touch and hit a low shot into the bottom corner which left the keeper a helpless spectator.
And from the very next attack, Arnold’s run towards the byline ended with a trip and another penalty.
Arnold himself sent the keeper the wrong way with the spot-kick for 3-2.
Melton looked in control until the 88th minute when Moore made a run into the area and found himself looking for support when he was crudely tripped.
The referee pointed to the spot for a third time, and Jamie Hall made no mistake from 12 yards.
Town co-manager Tom Manship quickly brought on sub Kadeem Price with the game going into injury time.
And with his first touch, Price won the ball in midfield and played a great ball over the defence for Lever who caught the ball on the half-volley and guided it past Farrell to send the large away support into a frenzy.
Melton host Burton Park on Saturday (ko 3pm) looking for a seventh straight win.
Town: Peet, Rigby, Gilchrist (Westwood), Hibbitt (Price), Wilson, Leon Fray, Arnold, Hill, Reeve, Parsons (Reek), Lever.