The home side were the better team from off and had the chance to take the lead after just nine minutes. A scramble after a corner saw Oliver Rathbone scuff his shot and send it straight to James McClean in a dangerous position, but his effort hit the bar.
Just six minutes later, the Red Dragons broke the deadlock as Ryan Barnett took on the ball and passed to Elliot Lee, who knocked it wide to McClean. His deadly ball into the net found Ollie Palmer perfectly placed to find the net.
Lee made sure that the home team went in at the break with a comfortable break when a long run from Barnett went unchallenged before he knocked it into the midfielder's path to roll into the bottom corner from the edge of the box.
It was thanks to some hapless Shrewsbury defending that Wrexham managed to knock home their third goal. A terrible clearance from Aristote Nsiala put his team-mates in a rough spot as goalkeeper Toby Savin came rushing out of his box, only for Jack Marriott to beat him to it, knock it past him and send it into the net.
There was no way for Shrewsbury to claw their way back into a game against a much stronger opponent, who now sit top of League One ahead of Stockport County, who have a game in hand.