An early own goal had put the visiting Salop side ahead, before Barnsley forward Sam Winnall equalised before half-time.
Clark struck a dramatic added time winner to spark mad scenes in the away end.
The visitors flew out of the blocks and broke at such a pace that no one was able to apply the finishing touch to Tyrone Barnett's appetising low cross.
Shortly after, skipper Liam Lawrence lofted a fine ball down the line to the galloping Tootle, but the wing-back cross was deflected behind - Abu Ogogo lifted the resulting header over the top.
The hosts first look at troubling Jayson Leutwiler was through the lively Conor Hourihane, the attacker feinted a shot before smacking a left-footed angled effort narrowly wide from 30 yards.
But within a blink Mellon's men were ahead, Lawrence's right-side corner was met by a crowd of players before squeezing over the line.
The officials were hesitant to award the goal, but it appeared Adam Davies in the Barnsley goal diverted Marc Roberts' touch into his own net.
Instead of the opener spurring Salop on, they rather crept back inside their shells and the hosts were allowed to dictate much of the play.
Arsenal loanee Dan Crowley impressed with some neat touches and curled over the top on 10 minutes.
Shrewsbury Town: Leutwiler, Tootle, Brown (Sadler 65), Ellis, Grandison, Knight-Percival, Lawrence (c), Ogogo, Wesolowski (Cole 15), Clark, Barnett (Akpa Akpro 76).
Substitutes not used: Halstead, Collins, Whalley, McAlinden.
Barnsley: Davies, Wabara, Roberts, Nyatanga, Mawson (c), Watkins, Scowen, Crowley (Harris 69) , Hourihane (Smith 69), Pearson, Winnall.
Substitutes not used: Townsend, Rothwell, Smith, Bree, Jackson.
Attendance: 8830 - 437 from Salop
Hourihane was picking up some dangerous positions and looked to be bearing down on Leutwiler, but fluffed his lines by dragging a shot wide.
Marley Watkins had a fine opportunity to draw the Tykes level but after cutting inside Nathaniel Knight-Percival, his low finish from inside the box was miraculously kept out down low by Leutwiler.
Despite still in front, Town suffered a blow as returning midfielder James Wesolowski was forced off with a knock to worried eyes from the Salop bench.
Hourihane was at it again and smacked Leutwiler's post with a 20 yard striker, the visiting glovesman did fantastically to divert the drive onto the woodwork.
But Leutwiler saved his best till the half hour mark. Watkins was released down the Barnsley goal but instead of taking a chance against the 'keeper, attempted to square to Winnall.
Mark Ellis' timely interception only sent the ball back to Watkins at the angle - who was denied by a remarkable sprawl from the Salop number one.
The hosts had their deserved leveller seven minutes before the break, there was nothing Leutwiler could do this time.
Winnall's fine touch deceived Jermaine Grandison before he slotted home a cool finish from the edge of the area.
Salop picked things up after the restarted and Clark's drive from distance looked to be finding the far corner, until Davies' intervention.
The officials awarded a goal kick - to the visiting supports' frustration. Clark was lively and his neat backheel set in Knight-Percival who wasted a decent crossing opportunity.
Larnell Cole was busy in the early stages of the second period and he set in Junior Brown, who did well to cut inside before flashing over.
Barnsley subs were having an impact and in particular Kadeem Harris, the left-sided attacker cut in but saw his shot well blocked from Lawrence.
The rebound fell for Josh Scowen who fired an effort into Knight-Percival - handball appeals waved away by referee Stuart Atwell.
But Mellon's final throw of the dice, Jean-Louis Akpa Akpro, passed up two glorious opportunities inside the final 10 minutes.
The forward inadvertently receives a rebound from Cole's cross and could only lash wide, before a peach of a delivery from Mat Sadler was nodded straight at Davies.
The lively Harris looked for a winner but fired over the top, as Barnsley struggled to break down a resolute Salop back-line.
Clark was heavily involved in the four minutes of added time, as first he dropped under pressure from Davies.
He then saw an prodded effort find the side netting before his wonderful winner sparked incredible scenes in the visiting end.
The attacker dribbled at the heart of the Barnsley backline and his strike took a deflection off a defender and beyond Davies, for Town's first league win of the season.

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