Leahy converted a penalty in the 89th in front of the 282 travelling Shrewsbury fans, after substitute Tom Bayliss was brought down in the box sending the fans into pandemonium behind the goal.
The Chairboys had taken the lead earlier from Garath McCleary's effort from distance after he collected a loose ball in midfield.
But Steve Cotterill's men hit back, equalising in the 60th minute after George Nurse launched the ball into the Wycombe penalty area, the throw was flicked on, and Pennington was on hand to head in at the back post.
But Leahy's penalty was enough to secure the points for Salop despite late pressure from Wycombe.
And the boss made two changes to the side that beat Carlisle United 3-2 in the Carabao Cup on Tuesday night.
They lined up in a 3-4-1-2 formation against The Chairboys, with the same back three of Mattew Pennington, Chey Dunkley and Tom Flanagan.
Julien Dacosta and Geroge Nurse played in the wingback roles. Captain Luke Leahy and Taylor Moore played as the two holding midfielders.
Jordan Shipley played in the number 10 role behind Daniel Udoh and Rekeil Pyke up front.
Shrewsbury were given a lift before kick-off when Wycombe's star striker Sam Vokes was not in the squad after he was ruled out with an injury.
Town really should have taken the lead within the first five minutes. They worked the ball well down the left, when Nurse played in Shipley he put in a great cross for Pyke, but from close range, he could not get his header on target. Then Dacosta recycled the ball back into the danger area but it was cleared by Wycombe.
It was a promising start for Shrewsbury, and they got in again down the left. Great work by Udoh to play in Nurse, who got to the byline and delivered a wonderful ball to the back post to Dacosta, but his diving header from a tight angle went miles over. Good chance though.
Salop looked like a threat on the counter attack.
But the hosts began to settle, and they grew into the game.
Initially, most of Wycombe's pressure came from set pieces. And when they finally got their delivery right, they should have scored. A free kick from the left was flicked on at the near post, and the free header was brilliantly tipped over by Marko Marosi.
Temperatures hit 34 degrees at times, so a drinks break was taken midway through the first half.
And when play resumed, Marosi was called into action again making a good save from an Obita strike.
The left-back shot from around 30 yards out, but it was beaten away by the keeper. The ball ricocheted, and Leahy threw his body on the line to block another effort from the rebound.
The Chairboys continued to probe. And they forced another acrobatic save out of Marosi, again from distance. Wing this time with the effort, which he hit powerfully, but the keeper dealt with the danger.
Wycombe looked threatening when they got the ball into the final third, with Mehmeti looking very lively down Shrewsbury's right.
And Town defended deeply but resolutely with Wycombe playing some attractive football and dominating possession. It was always going to be a tough afternoon at Adams Park for Town, but Cotterill would have been happy at the break.
And five minutes into the second half Wycombe took the lead. The ball was bouncing around inside the Town half. It was headed up in the air by Flanagan. McCleary picked up the loose ball and unleashed a shot from 25 yards out into the bottom corner.
McCleary, was a threat all afternoon for Wycombe and the goal gave him confidence. Shortly after, he tried his luck from distance. But this time Marosi was equal to it. An important save to keep the score at 1-0.
And on 60 minutes Town scored their first league goal of the season after Pennington nodded in from a long throw. Nurse launched the ball into Wycombe's box, Flanagan won the first header, and Pennington was on hand at the back post to head the ball back across goal and into the corner.
The defender celebrated in front of the delighted Shrewsbury Town fans.
Wycombe continued to have a lot of the ball, but Shrewsbury were resilient, and they kept The Chairboys at bay.
Tom Bayliss came on for Moore, to provide Shrewsbury with extra energy given the extreme weather conditions.
And in the 89th minute, Tom Bayliss was hauled to the ground after getting the wrong side of the defender, and referee Lee Swabey rightly pointed to the spot.
Captain Luke Leahy converted the penalty with a powerful strike straight down the middle. With the Salop players celebrating in front of the travelling Shrewsbury fans.
Wycombe pushed hard for an equaliser with 7 minutes of added time.
But despite a few half chances, Shrewsbury stood firm.
Salop: (3-4-1-2): Marosi, Pennington, Dunkley, Flanagan, Dacosta (81mins), Nurse, Moore (Bayliss 66mins), Leahy, Shipley, Pyke, Udoh (Bloxham 86)
Subs: Burgoyne, Bowman, Bloxham, Caton, Bayliss, Craig, Barlow.
Wycombe: (4-2-3-1): Dickinson, Grimmer, Jacobson, Forino, Wing (Freeman 77), Mehmeti, McCleary, Horgan (Young 77mins), Mellor (Wheeler 85mins), Obita (Mawson 85mins), Scowen.
Subs: Blunkell, Wheeler, Young, Mawson, Freeman, Al-Hamadi, Pattenden