James Collins netted his second hat-trick of the season, with Luke Berry netting his first goal since his return from injury to help see off a Posh side that played with ten men for three-quarters of the game following Ryan Tafazolli's 25th- minute sending off.
But that point the Town were already a goal to the good, with Collins netting his 12th of the season. Number 13 followed two minutes after the straight red, for a foul on Andrew Shinnie, before 14 arrived on 53 minutes.
Berry rounded off a superb first full game for himself with the fourth as interim manager Mick Harford - who signalled to the crowd at the end that it was the players who should take their applause for a stunning performance - oversaw a 15th successive league game without defeat.
Harford made one change to his side following the midweek FA Cup exit, when the Town had followed up the 1-1 draw at Sunderland in his first game back at the helm with another impressive showing, without reward, in the 1-0 defeat to Championship club Sheffield Wednesday.
Returning to the starting line-up after being rested on Tuesday was Alan McCormack, with Elliot Lee missing out with a hamstring injury.
New signing from Barnsley, George Moncur, started on the bench, along with fellow January signing on loan from Derby, George Thorne.
So the Hatters lined up with James Shea in goal, behind a back four of Jack Stacey, Matty Pearson, captain Sonny Bradley and James Justin. McCormack, Pelly-Ruddock Mpanzu, Berry and Andrew Shinnie, with Collins and Kazenga LuaLua up front.
The Town madepositive a start and had their noses in front by the eighth minute when Shinnie was given the freedom of the midfield to play Stacey to the right byline, and Collins was there inside the six-yard box to turn it over the line.
Cardiff City loanee Lee Tomlin fired wide on 17 minutes after Stacey gave the ball away to Kyle Dempsey, then Shea had to get down low to keep out a Marcus Maddison shot from outside the box, before the ex-Gateshead man hit a free-kick over the bar.
The visitors were reduced to ten men in the 25th minute when centre-half Ryan Tafazolli was shown a straight red by referee David Coote for a dangerous looking challenge on Shinnie.
Within two minutes the Hatter had doubled their advantage and it was the same trio of players involved, Shinnie playing Stacey inside Dembele on the Town right, with the full-back cutting back a lovely cross for Collins to turn in with a brilliant finish at the near post.
Berry was next to have a go, forcing Posh keepeer Aaron Chapman into a full-length save from 25 yards, before the high press from Shinnie and LuaLua won the ball for Mpanzu to fire at goal, but he dragged his shot well wide.
Collins was presented with a glorious chance for his hat-trick in the 36th minute when Shinnie played Justin in on the left, and the full-back's first-time cross was cleared only as far as the Town's top scorer, who couldn't direct his left-foot volley on target.
LuaLua extended Chapman on 39 minutes, but the Posh keeper was equal to his low shot, then McCormack shot wide at the end of a move he'd started by winning the ball and creating space with a brilliant Maradona turn, before playing Shinnie on the left to cross low for Collins, who didn't connect properly with his shot.
Posh had already made one substitution in the first half, bringing Joe Ward on for Dembele in the immediate aftermath of the Town's second goal, and they completed their three changes at the break.
Tomlin and Dempsey were withdrawn, replaced by George Cooper and Alex Woodyard, but it made little difference to Peterborough's hopes.
Collins completed his treble with the best of the lot eight minutes after the interval, LuaLua nicking the ball ahead of Danny Lafferty on halfway and sprinting down the right before cutting inside and slipping a lovely ball into Collins' path.
The ex-Crawley man didn't need to break stride as he buried his left-foot shot into the bottom corner, beyond Chapman's outstretched right arm.
Town were rampant and it was four in the 66th minute when LuaLua was the architect again, this time wriggling his way around a defender to play Berry into the area. The midfielder made no mistake from 16 yards, swivelling and passing his effort around Chapman and into the beck of the net.
Harford introduced the two new Georges - Moncur and Thorne - immediately after the goal and Moncur almost had a brilliant debut goal in the 77th minute. The midfielder had already produced one backheel in a patient passing move dictated by Thorne, when he collected the ball 30 yards from goal.
He traded passes with LuaLua, who returned it with a backheel, then waltzed past two defenders before shooting past Chapman, only to be denied by Ben White, who had raced back to slide in and clear off the line.
Maddison put the ball in the back of the net at the other end, but the flag was long since up for offside, then he fired high over the bar inside the last five minutes.
LuaLua, desperate for a goal his performance deserved, did likewise in front of the Oak Road, then Moncur bent one just over before Sheehan saw a 30-yarder deflected behind for a corner.
It didn't matter, the Hatters already had enough in the bag to move within a point of Pompey, who lost at Sunderland, and three points clear of Sunderland, who drew at Scunthorpe.
Onto Southend next Saturday!
TOWN: Shea, Stacey, Pearson (Sheehan 77), Bradley (c), Justin, McCormack (Thorne 69), Mpanzu, Berry, Shinnie (Moncur 69), Collins, LuaLua. Subs: Potts, Jarvis, Jones, Isted.
Goals: Collins 8, 27, 53, Berry 66
Yellows: Pearson
POSH: Chapman, Naismith, Tafazolli, White, Dembele (Ward 28), Cooke, Toney, Lafferty, Maddison, Tomlin (Cooper 46), Dempsey (Woodyard 46). Subs: Godden, Reed, O'Malley (GK), Stevens.
Yellows: Maddison
Red: Tafazolli
REFEREE: David Coote
