It looked plain sailing for the home side as they established a 3-0 lead just after the break, but Stockport hit back with two Anthony Pilkington goals to set up a tense finish.
The outcome was in doubt until the dying seconds when Cooke completed his hat-trick to ensure a Shrewsbury victory. There was even more misery to follow for the visitors as central defender Ashley Williams was ordered off right at the death for a professional foul.
After a fine win at Darlington last Saturday following two straight defeats, Shrewsbury were quickly out of the blocks and were ahead on four minutes with a fine individual goal from the lively Ben Davies.
Cooke struck for the first time on 21 minutes and put Shrewsbury three goals to the good in the 48th minute.
Pilkington netted a minute later and then brought Stockport right back into contention with his second in the 69th minute - with the Shrewsbury goal often living a charmed life before Cooke finally put the issue beyond all doubt.
In the fourth minute Shrewsbury swept down the pitch with Davies put clear of the visitors defence. Although he was momentarily halted, Davies retained enough composure to slide the ball beneath James Spencer to give the home side an early breakthrough.
With a quagmire pitch making life difficult for defenders, there were near things at each end before Shrewsbury struck again on 21 minutes. At first it seemed Davies would score again, only for his drive to be blocked on the line - but the rebound fell nicely for Cooke who smashed it home from close range.
Three minutes after the restart Spencer was beaten again - and left stranded as Cooke seized on a long throughball before scoring with a delightful chip from 20 yards.
But Stockport, who themselves came into the match on a good run of form that has seem them rapidly climb the table, quickly rallied and a well-taken goal from the impressive Pilkington, who scored with a neat curling shot to reduce the arrears.
In the 69th minute they cut the deficit even further when Pilkington nodded home a second and throughout the remainder of the match seemed destined to snatch a draw, with substitute Michael Malcolm going closest when his effort struck a post before being hacked to safety.
But all their efforts were to be in vain when Cooke struck again in injury time, completing a fine individual display by side-stepping the advancing Spencer and netting his ninth goal of the campaign from a tight angle - as Shrewsbury hoisted themselves to within three points of a play-off place.