Jordan Clark netted a superb solo goal to give the Hatters the lead just after the half-hour mark in an even first half.
But Boro, 15th at kick-off but unbeaten in four under their own new manager in Michael Carrick, levelled soon after through Chuba Akpom's header.
The Hatters were reduced to ten when Amari'i Bell saw red for a second yellow card with just under 20 minutes plus injury-time remaining - both for fouls on Boro's right-sided flyer Isaiah Jones - and despite some stubborn resistance, substitute Matt Crooks fired in the hosts' winner a minute into added time.
Edwards named an unchanged line-up from the one that interim manager Mick Harford selected for the 1-1 draw with Rotherham a month ago, with Tom Lockyer keeping the captain's armband in Sonny Bradley's absence.
Ethan Horvath returned to training on Thursday morning and was straight back in the starting line-up following USA's World Cup campaign, whilst there is a single change on the bench with Cameron Jerome replacing Josh Williams.
The Hatters lined up in a familiar way with Gabe Osho, Lockyer and Amari'i Bell as the three centre-halves, James Bree and Alfie Doughty as the wing-backs either side of the midfield trio of Pelly-Ruddock Mpanzu, Clark and Allan Campbell. Top scorer Carlton Morris and Elijah Adebayo led the line.
After Tommy Smith had blasted Boro's first shot way over in the fifth minute, Doughty was the first Hatter to have a sight of goal seconds later, but his left footer flew narrowly over Zack Steffen's bar.
Town were looking comfortable in an even game, with Doughty continuing his promising form from before the World Cup break down the left. But it was Bell, just behind him, who was the first name into ref Gavin Ward's book for a foul on Isaiah Jones in the 25th minute.
The breakthrough came in the 33rd minute when Clark picked himself up after being fouled by Jones just inside the Boro half, and took a quick free-kick into Morris feet.
Receiving the ball back from his front man, Clark drifted to his right, beat Jonny Howson with a nutmeg, then Hayden Hackney with a drop of the shoulder before unleashing a left-foot bullet past Steffen into the bottom corner.
The lead lasted for just five minutes as Boro, who had seen Jones fire over straight after the goal, moved the ball from right to left across the edge of the Town box, with Ryan Giles eventually delivering the cross for Akpom to meet with a firm downward header past Horvath.
Level at the break, the Hatters started the second half on top and Lockyer was presented with a glorious chance to restore the lead three minutes after the restart, but the man returning from World Cup duty with Wales volleyed Doughty's deep free-kick wide.
Lockyer picked up a 57th minute booking for a trip on Jones, who had appeared to kick out at the defender on the right touchline seconds earlier, causing the fourth official Leigh Doughty to intervene.
The right sided flyer was proving to be the hosts' main threat, but Bell and Doughty were shackling him well on the Town left.
It was Doughty's pressing at the other end that helped create an unexpected chance for the Hatters on 64 minutes, with Middlesbrough keeper Steffen closed down by Adebayo and Campbell after Doughty had initially nicked the ball on the left flank.
The ricochet off the Scottish midfielder went into the side-netting, just inches the wrong side of the post as Steffen was hounded on the edge of his six-yard box attempting to play out.
Edwards brought Harry Cornick and Luke Freeman on for Adebayo and Campbell, but Town were reduced to ten men with just under 20 minutes left when Bell picked up a second yellow for a foul on Jones.
The Hatters reverted to two banks of four, with Doughty dropping in at left-back after receiving treatment, and continued to stand firm as Lockyer headed one deep cross behind, then blocked an effort from distance before Boro sub Duncan Watmore drilled a low shot in that Horvath held well with ten minutes to go.
Howson and sub Crooks wasted good chances as the clock ticked down, but just as the board went up for five minutes of added time, the latter steered in a volley from close range to take all three points for Boro and lift them above Town in the table.
Goals:
LT - Clark 33
M - Akpom 38, Crooks 90+1
Att: 23,187 (463 Hatters)
