Leon Clarke headed home the winning goal in the fifth minute, and United thoroughly deserved the victory as they held on for an important three points.
The Iron entered the game making three changes from the side which drew with Sheffield United a week earlier.
Jimmy Ryan came in for Andy Barcham in midfield, while new strike duo Karl Hawley and Leon Clarke replaced Mike Grella and Connor Jennings in attack.
Sam Slocombe started in goal, with Christian Ribeiro, David Mirfin and Tom Newey forming the defence.
The midfield consisted of Ryan, David Prutton, Josh Walker and Mark Duffy, while Clarke and Hawley led the attack.
The substitutes were Barcham, Grella, Jennings, James Severn, Robbie Gibbons, Jordan Keegan and new addition Chris Sedgwick.
Wearing all white, the away side started the game at a high tempo and Ryan advanced down the right on three minutes, sending a right-wing cross which goalkeeper Chris Weale palmed away and saw subsequently cleared.
However, just two minutes later. Scunthorpe went ahead as Clarke guided home a close range header in front of a delirious away end after a right-wing cross from Hawley, who had done well to hold the ball up and clip in a decent ball for his strike partner, who made it two goals in two games for the club.
Shrewsbury pushed for a leveller after that goal but the next chance came the Iron's way, as Duffy received the ball on the right hand side and drifted inside before firing a low left-footed effort just wide.
On 11 minutes, the Iron's bright start continued as Hawley sent a header just over the crossbar following Duffy's right-wing cross.
Five minutes later, there were shouts of handball from the hosts as Ribeiro blocked Mark Wright's header after Paul Parry's cross. The referee was having none of it though and the Iron played on.
The home side had their best chance so far midway through the first half as Tom Bradshaw capitalised on a defensive lapse and advanced towards Slocombe dragged his effort across, and just wide of, the goal.
Then, shortly after, Jermaine Grandison scrambled an effort wide for the hosts.
Seconds later, Duffy almost made it two as he skilfully found space in the area before lashing a left-footed drive just wide of goal.
In the 30th minute he came close again as he had a close range shot saved on the line by Chris Weale's left leg after good play by Hawley on the right to set him up.
Scunthorpe remained a big attacking threat and, on 37 minutes, good play on the edge of the area saw Clarke clip through Walker but Weale was quick to advance and stop the attack.
After more decent United pressure, the last chance of the half came Bradshaw's way in the second minute of stoppage time but he stumbled on the ball and the visitors cleared their lines.
Two minutes into the second half, Hawley forced a save from Weale, as his scrambled effort at the near post was touched behind for a corner, from which nothing came.
In the 49th minute, Grandison drew a save out of Slocombe at his near post, as he played a one-two down the right side of the area and got into space before unleashing a shot.
Four minutes later, Canavan scrambled a low shot into Weale's arms after Clarke's flick down from a Prutton corner had reached him to twist and turn.
Receiving from Luke Summerfield, who cut inside, Wright drifted in from the left and curled a right-footed strike just wide of goal in the 56th minute as the hosts remained a threat.
Just a minute later though, Scunthorpe almost made it two, as Clarke received the ball from Hawley and fired past the goalkeeper from close range, but Darren Jones blocked it on the line.
On 59 minutes, good trickery from Duffy eventually saw his effort from the right deflected wide, and then Ryan struck an effort wide of goal following the subsequent corner by Prutton.
The next chance arrived Shrewsbury's way six minutes later, as Parry's free-kick from the right took a deflection over the crossbar as the Shrews seeked a leveller.
The 67th minute almost saw a wonder goal from Ryan, as he showed great improvisation to turn and fizz a thunderous left-footed volley just wide of goal from 25 yards.
Ten minutes later, following a lull in play, the Iron broke forward through Hawley, whose curled effort from 25 yards drew a routine save from Weale.
As the game entered the final five minutes, Scunthorpe forced a succession of corners through Newey crosses which were tipped away, and then a Prutton drive which was deflected wide.
Jennings then came on for Hawley as United made their first change of the afternoon.
There were four minutes of stoppage time given, and Matt Richards drilled a free-kick over the crossbar for the hosts from 25 yards.
Then, a minute later, Sedgwick came on for his debut to replace Ryan, just before Jennings found space in the area and saw his low powerful drive stopped by Weale.
Thankfully, there was no more pressure and Scunthorpe held on for victory, ending a 34 game unbeaten record for Shrewsbury at the New Meadow.
The Iron are next in action on Tuesday night when they visit Oldham Athletic.
Shrewsbury Town: Weale, Grandison, Summerfield, Jones, Purdie (Wildig, 68), Wright (Doble, 86), Richards, Bradshaw (Hall, 68), Morgan, Hazell, Parry.
Substitutes: Anyon, Jacobson, Hector, Gornell.
Scunthorpe United: Slocombe, Ribeiro, Mirfin, Canavan, Newey, Ryan (Sedgwick, 90), Prutton, Walker, Duffy, Clarke, Hawley (Jennings, 86).
Substitutes: Severn, Grella, Barcham, Gibbons, Keegan.
Referee: Graham Salisbury.
Attendance: 5,623 (278 away).