Shaun Whalley opened the scoring on his return to the side with a deflected strike before the Town winger added a second and Nathanael Ogbeta rounded off the victory with a stunning third.
It was an Easter afternoon to remember for Cotterill's side, their biggest league victory of the season and their biggest in League One since October 2017.
And the players received a deserved ovation from their manager, 100 miles away in Bristol. Assistant Aaron Wilbraham said: "The gaffer was pleased, when we first rang you just heard clapping down the speaker and everyone realised he was giving a round of applause.
"It makes his day when he gets to see the lads each week, it's the only time he gets to see them and when they do him proud like that it makes his day."
Wilbraham revealed he felt a steely determination in Town players over the Bank Holiday weekend to end a run of two defeats.
Shrewsbury remain 17th in League One but climbed to 49 points and now look all-but certain to have secured safety, with games in hand on rivals above them.
He added: "I thought the boys had a determination, since the game on Friday against Northampton.
"They came in over the weekend with a real focus in their eyes, I could see it in the game today, it was a good performance and the 3-0 win was great for everyone involved.
"A lot of our wins have been 1-0 but it's nice to score three goals and enjoy the latter stages but still keep the concentration and focus for a clean sheet."
Wilbraham said of the contentious opening goal, which has been credited in some places as a Will Aimson own goal: "It was definitely a brace for Shauny, it was great to see him back among the goals and it was a tremendous goal from Nathanael as well."
Town named Aaron Pierre on the teamsheet only for the defender to be replaced by Ro-Shaun Williams as a last-minute precaution due to concerns over the defender's calf.
"It was precaution with Aaron's calf, not the one he was injured with, maybe it was a little fatigue from Friday but after what happened last time with Aaron we didn't want to chance anything," Wilbraham added.
"It was a good decision in the end, regardless of the result, to pull him out at the late stage. It was important not to make anything worse.
"And it shows what a professional Ro-Shaun is coming in to play as well as he did after being told at the last minute at 10 to three, he switched on and put in a performance like that."