The home side caught the first clear sight of goal on six minutes. Paul Tierney played in Adam Proudlock, but Shrews defender Kelvin Langmead raced in and produced a splendid last-ditch tackle.
County's Dominic Blizzard saw a shot blocked, before young Irishman Gary Dicker fired wide from all of 25 yards.
Stockport had started well and, on 15 minutes, Proudlock fizzed in an angled drive which went narrowly wide.
At the other end, Shrewsbury responded when Dave Hunt blazed over from just outside the box, before they then took the lead on 25 minutes.
Marc Pugh strolled down the left and crossed and there was Dave Hibbert who rose well to head home into the corner.
Scorer Hibbert flew in to plant another header just the crossbar seconds later, but Stockport went on to level matters just beyond the half-hour mark.
Matty McNeil's low drive was bravely blocked, only for Proudlock to react quickest as he planted home the rebound from 12 yards.
Stockport's Anthony Pilkington was having a good half and he fired in a great shot which went inches wide just before the break.
Shrewsbury had the first chance of the second half, but skipper Stewart Drummond's shot was blocked by Blizzard.
Michael Symes then saw an effort deflected wide for a corner-kick by Ashley Williams.
At the other end, Stockport substitute Anthony Elding just failed to connect with a great cross from Proudlock.