The striker's curled set-piece in the 38th minute fired Town into the Third Round of the Capital One Cup and means the Club are still unbeaten in the 2014/15 campaign.
There were chances for Town to win by even more on the night after hitting the post three times, Leicester once at the other end, but it was not to matter in the end and the Shrews will enter the draw for Round 3.
Manager Micky Mellon made three changes to the side which drew 1-1 with ten men at Northampton Town on Saturday - two forced upon him.
Jermaine Grandison, Ryan Woods and James Collins all came in for Jean-Louis Akpa Akpro, the injured Ashley Vincent and the suspended Liam Lawrence.
The two teams came out to a rapturous applause at the King Power Stadium, which hosts Premier League football this season, with Town greeted by 1,121-strong army of Shrews supporters.
But it was Leicester who had the first shot on goal, as Jayson Leutwiler had to flick out a leg to turn behind Riyad Mahrez’s low drive across goal from the right in the opening minute.
Town came back at the other end, though, as James Collins headed just over inside the box from Jordan Clark’s free-kick on the left - Town’s first threat of the match.
Both teams also tried their luck from long range. Paul Koncheskey fired straight into the hands of Town’s Swiss goalkeeper Leutwiler while, at the other end, Clark clipped over a cross from the right and it caught Ben Hamer off guard, touching the far post on its way past the goalkeeper.
Defender Micky Demetriou then blasted an effort from outside the box and forced Hamer to tip over.
When Leicester next went forward, Mahrez was booked for taking a tumble inside the box in the 24th minute.
The Foxes had great chances of their own as well. Marcin Wasilewski forced a superb reaction save from Leutwiler from close range before he lashed his second attempt over.
Two minutes later, Chris Wood headed against the Town crossbar from Anthony Knockaert's free-kick on the left, with Connor Goldson eventually clearing the danger.
At the other end, Woods’ hung a ball in from deep but Grandison nodded his free header in the box well wide.
Goldson also attempted to catch Hamer off his line with a quick free-kick ten minutes before half-time, but it flew safely over to the relief of the home supporters.
However, in the 38th minute, Mangan and Demetriou stood over a free-kick on the edge of the box before the former curled the ball away from Hamer and into the left corner to give Town the lead at the King Power Stadium.
Unfortunately, the striker pulled up with a hamstring injury in the moment of his goal and had to be replaced by striker Scott Vernon.
Town could have gone in ahead by more than the one goal, though, as Demetriou skewed wide a shot from the left when teed up by the substitute Vernon.
Former Leicester man James Wesolowski also drove wide on the run from the left, but Town were well worthy of their lead at the break.
HALF-TIME: Leicester 0-1 Town
As expected, the Premier League side looked to up the pressure on Town in the second period and find an equaliser.
David Nugent and Andy King were both brought on to freshen up the attack in Nigel Pearson’s side.
But Town remained resilient in defence and, at the other end, Goldson got his head on a Clark corner but could not direct it on target.
Town should have wrapped the game up with 15 minutes left as well. Demetriou sent a crossfield ball up to Collins who chested it down before unleashing a dipping half volley which crashed back off the post.
The rebound fell kindly for Vernon just outside the six-yard box, but he too could only prod his effort back against the right post with just the goalkeeper to beat.
It was an agonising moment for the Shrews fans located by the corner of the goal Town attacked in the second half, but they had their hearts in the mouths at the other end moments later.
Leicester’s Knockeart battled his way to the byline before eventually pulling ball back for substitute Nugent, and the striker placed his effort towards the left corner but it too crashed back off the woodwork.
Micky brought on Andy Robinson and Akpa Akpro with five minutes to go, looking to see the game out, while Woods tried his luck from long range as Town sprung on the counter attack late on but fired over.
The referee added four minutes as the Foxes looked to fight back and Koncheskey’s blast over from long range was the closest they came.
But none of Town’s misses were made to matter in the end, though, as they pulled off one of the biggest upsets in the Club’s history.
Shrews will now join the likes of Manchester City, Liverpool, Arsenal and Chelsea in the Third Round draw.
FULL-TIME: Leicester 0-1 Town
TOWN LINE-UP: Leutwiler, Grandison, Demetriou, Woods, Goldson (c), Collins (Akpa Akpro, 84), Gayle, Wesolowski, Mangan (Vernon, 40), Knight-Percival, Clark (Robinson, 84).
SUBS: Halstead (GK), Smith, Wildig, Caton, Robinson, Vernon, Akpa Akpro.
BOOKED: Woods, Gayle.
GOALS: Mangan (38).
LEICESTER LINE-UP: Hamer, Koncheskey (c), Moore, Taylor-Fletcher, Knockaert, Mahrez (Schlupp 69), Wasilewski, Pearson, Cain (King, 57), Wood, Watson (Nugent, 57).
SUBS: Smith, Hopper, Schlupp, Barmby, Morgan, King, Nugent.
ATTENDANCE: 8,017 (1,121)