The visitors stunned Shrewsbury by taking a two-goal lead inside the first 20 minutes.
Pierce Sweeney got the first from a corner, and Jack Sparkes' 30-yard effort took an unlucky deflection off Jonathan Grounds to stun the Meadow into silence.
Town have shown resilience at times this season, and they did it again going into the break level after Matthew Pennington netted the equaliser and Tom Bayliss got his first goal for the club.
The chances continued into the second half with Nurse hitting the bar for Town, and Sam Nombe forcing Marko Marosi into a very good stop.
But Chey Dunkley won it a right at the death for Shrews when he smashed home a George Nurse long throw.
Steve Cotterill made one change to the team that beat Forest Green Rovers just 10 days ago.
Rob Street came in for his first start for Town and his home debut at Montgomery Waters Meadow.
Matthew Pennington, Chey Dunkley and Tom Flanagan – who have been ever present this season – made up the back three, with George Nurse and Rekeil Pyke on either side of them.
Skipper Luke Leahy continued to partner Taylor Moore in midfield.
And Jordan Shipley and Tom Bayliss supported Rob Street, who was the lone striker.
Prior to kick-off, players coaches and fans paid tribute to Her Majesty The Queen, who passed away last week after more than 70 years on the throne.
The players made their way on to the pitch and observed a minute's silence, followed by the national anthem.
The tributes were observed respectfully by everyone in the ground – a fitting tribute to Britain's longest-serving monarch.
Exeter started the game brightly, passing the ball around well. It took Town a little while to settle down and they were guilty of giving up possession a little too much in the opening exchanges.
The Grecians took advantage and took the lead through their captain Sweeney after Street was beaten in the air at the back post. The ball fell to the defender, who lashed it into the back of Marosi's net.
Moments later, the visitors got a huge slice of luck when they doubled their lead. Sparkes hit a powerful strike from a distance that was going wide, but it hit Grounds, completely wrong-footing Marosi and going in off the post.
The crazy first half did not stop there, and Cotterill's men hit back immediately when Pennington bundled the ball over the line following a scramble from a corner.
It was end-to-end, and both sides could have scored more goals, strange considering they were two of the league's tightest defences.
Sparkes could have made it three for Exeter when his header at the back post went wide, and they were made to pay for that miss.
In the 34th minute, Shrews were on terms when Bayliss got his first goal for the club.
A corner was cleared by Exeter, but Pyke hooked the ball back into the box, Dunkley won the flick-on and Bayliss was there to fire home.
Exeter were good on the ball and they caused Town problems, with Jevani Brown being particularly bright for the Devon-based side.
And the forward hit the bar in the with a lovely strike from 30 yards out.
At the start of the second period, the tempo of the game relaxed a little and Salop had more of the ball.
Bayliss continued to look good, and he was Town's most threatening player. He picked up a loose ball, carried it to the edge of the area, and almost found the corner. But his shot was tipped behind by the keeper.
Exeter had their opportunities to retake the lead and their best fell to Nombe, who produced a good save from Marosi.
He got in down Town's left and shot across the keeper who got a strong hand to it and palmed it away.
Town thought they had won it near the end when Bayliss's cross was missed by Ryan Bowman but fell kindly to Nurse. The left-back hit the ball first time, getting a lovely connection with the ball, but it rattled the crossbar and went behind.
Cotterill's men pushed for a winner as Saydee saw an effort blocked behind and Dunkley headed over from a corner.
But Town won it right at the death when Dunkley hammered the ball home from a Nurse long throw to spark scenes of jubilation in the stands for the home fans.
Salop: Marosi, Flanagan, Dunkley, Pennington, Nurse, Moore, Leahy, Pyke, Bayliss (Winchester 92), Shipley (Bowman 66), Street (Saydee 66).
Subs: Burgoyne, Bowman, Bloxham, Winchester, Caton, Saydee, Craig.
Exeter: Blackman, Caprice, Sparkes (Stansfield 71), Harper (Hartridge 71), Collins, Nombe, Key, Brown, Sweeney, Grounds.
Subs: Brown, Hartridge, Jay, Stansfield, Smith, Chauke, Cox.