Smith struck in the 8th, 48th and 81st minutes to seal victory for the Owls, although it wasn’t enough to keep their hopes of automatic promotion alive with wins for Ipswich and Plymouth elsewhere.
For Shrewsbury, the scoreline was perhaps harsh as they put in a performance that certainly didn’t have an end-of-season feel, battling away and enjoying several spells of decent pressure without being able to find a goal.
And ultimately, it was a difference in clinical quality from the visitors which saw them pick up a deserved win, Smith’s hat trick just three of many decent chances which gave Marko Marosi a busy afternoon in the Shrewsbury goal.
It’s now eight games without a win for the injury-hit Shrews who will be looking to round off their home programme on a more positive note when Bristol Rovers are the visitors on Tuesday.
Steve Cotterill made three changes to the Shrewsbury line-up, with Christian Saydee back in after coming off the bench at Bolton in midweek following injury, Elliott Bennett also in the starting eleven and Tom Flanagan returning from suspension.
That trio replaced Matt Pennington, the matchwinner in last season’s corresponding fixture, who was missing from the squad, and Tom Bloxham and Ryan Bowman, who both dropped to the bench.
The Owls, needing a win at the start of the day to have any chance of automatic promotion from League One, included former Town loanee Dominic Iorfa in their starting eleven.
The rain was teeming down at Montgomery Water Meadows for kick-off, but there was a lively atmosphere around, the Shrewsbury fans responding in kind to the noise being made by the packed ranks of Wednesday fans at the opposite end of the ground.
Some neat interplay between Street and Luke Leahy sent Bennett away down the right but his cross went behind as Town went in search of a positive start.
The Owls first opening came on six minutes as Marvin Johnson’s cross was met with a header from Smith which comfortably cleared the crossbar.
And it was Smith who broke the deadlock just two minutes later after a neat move which carved open the Shrewsbury defence.
Iorfa brought the ball out of defence to find Smith, who pass to Gregory was returned with a backheel allowing the goalscorer to stride on and beat Marosi with a clinical finish.
That breakthrough certainly did nothing to quell the noise coming from the Wednesday fans but Shrewsbury went for a fairly quickfire response as Carl Winchester won the ball in midfield and fed Street, whose low drive was comfortable for Cameron Dawson.
But it was a brief respite as the pumped-up Owls pressed forward again.
Chey Dunkley was needed to make a hurried clearance after Johnson flashed a ball across goal and Marosi was then called into action to tip a rising drive from Barry Bannan over the crossbar.
It was taking plenty of battling defending from the Shrews to try and keep the visitors quiet and Killian Phillips flew into a block to deny Will Vaulks after the ball had sat up invitingly just outside the area.
But having appeared to weather something of a Wednesday storm Town enjoyed a spot of possession in their opponents’ half. Jordan Shipley did well to win the ball and find Saydee whose chip into the area was met by Street with a shot deflected wide for a corner. That proved the prelude to a string of Town half chances with crosses going into the box and shots from Shipley and Saydee blocked by what was now a far more stretched Owls defence.
Bennett was whipping in some excellent deliveries from the right flank and from a free kick a clearance dropped to Shipley for a difficult half volley chance which flew over.
Wednesday however were still looking a threat when going forward, and a spot of pinball which seemed to last eternity on 38 minutes somehow ended with Smith directing his effort wide.
That was the last action of note from a first half in which Shrewsbury enjoyed one particularly prominent spell but were finding it a challenge to keep Wednesday quiet at the other end of the pitch.
Cotterill opted for a change at half time, with Bloxham coming on to replace Saydee.
Within 30 seconds of the restart the Owls went so close to extending their lead.
Gregory’s shot from a pass from Dennis Adeniran was diverted inches wide of the target by Flanagan.
But they didn’t have to wait long to extend their lead, with Smith adding his second on 48 minutes.
It was another fine move which was finished with a well-placed header from Smith after meeting Johnson’s cross from the left.
Town didn’t let their heads drop however and probably should have pulled one back in the 51st minute.
Leahy and Shipley were involved in teeing up Phillips who couldn’t get enough purchase on his first time shot which was easily gathered by Dawson.
And the Owls were still flexing their muscles in attack with Marosi again called into action to palm away a powerful first time strike from Vaulks.
There was then a bizarre moment as Owls sub Aden Flint who was warming up and appeared to get in the way of Bennett placing the ball in the quadrant for a corner. As Bennett complained to referee Dean Whitestone, the official’s response was to come over and show the Shrewsbury man a yellow card.
Town kept up the pressure and Dunkley did well to get his head on a deep cross from Shipley to force Dawson into a save.
Approaching the midway point of the second half came a further two Shrewsbury changes, with Bowman and Rekeil Pyke on for Street and Bennett.
And the hosts were still not giving up the ghost as they tried to find a way back into the game.
A well-executed move saw the ball worked to Leahy just outside the box and his precise left foot effort just glanced the top of the crossbar with Dawson flailing.
Back at the other end Marosi had to be alert to parry a drive from Vaulks who tried his luck with a low free kick around the wall.
But with ten minutes remaining the Owls made it three as he tucked the ball home after a challenge on Flanagan which the Shrewsbury defender insisted was a foul.
Referee Whitestone was having nothing of it though and dismissed the protests, instead showing Flanagan a yellow card, as Wednesday moved into a three-goal lead.
Bloxham went in search of a consolation for Town three minutes from time but his shot from the angle was high and wide.
Town were still pressing forward in added time to try and find a crumb of consolation but were unable to break down a resilient Owls back-line.
Town: Marosi, Moore, Dunkley, Flanagan; Bennett (Pyke 66), Winchester, Leahy, Shipley; Phillips; Street (Bowman 66), Saydee (Bloxham 45).
Subs: Burgoyne, Craig, Barlow.
Wednesday: Dawson, Iorfa, Ihiekwe (Flint 75), Palmer (Brown 88), Adeniran (Bakinson 75), Vaulks, Bannan (Windass 89), Johnson, Paterson, Gregory, Smith (Dele-Bashiru 84).