The winger struck the only goal of the game in stunning style in the 64th minute - only two minutes after he had been introduced from bench by Posh manager Darren Ferguson.
Mendez-Laing collected a pass from skipper Tommy Rowe, cut inside from the left flank and steered a 20-yard shot past goalkeeper Chris Weale.
The goal will have come as a major source of relief for Peterborough, who were well short of their best after Paul Taylor had hit a post during an early flourish.
Tyrone Barnett and Michael Bostwick both failed to make the most of free headers from Danny Swanson corners but Posh rarely came under threat from a Shrewsbury side yet to win on the road this season.
The visitors' best chance was spurned by substitute Liam McAlindon when he dragged a shot tamely wide after seizing on a poor pass from home defender Craig Alcock.
Posh goalkeeper Bobby Olejnik was not required to make a noteworthy save until keeping out a Jon Taylor effort in the closing stages.