Town picked up a deserved three points but found themselves on the back foot once again at Montgomery Waters Meadow after a short free-kick played in Wimbledon territory backfired and the Dons broke to open the scoring through Dapo Mebude.
Shrewsbury did, however, get in level at the break as Luke Leahy - one of those involved in the earlier set-piece breakdown, atoned with his first goal for the club with a looping header from an Elliott Bennett cross, the latter who lost the ball for Dons to register.
The second period did not prove as lively as the 45 minutes that went before it but Town snatched what felt like a significant victory to lift them out of the League One drop zone to 20th as Daniel Udoh emerged as a substitute to convert a fine header for his first goal of the season.
The manager revealed his players' short set-piece routine was not one worked on in training, but was willing to forgive them for an otherwise hard-working first half display. He did, however, call for his players to not go 'off script'.
Cotterill said: "It was very important (to score quickly after opener).
"It's also very important, when we're in those areas, to do what we've done on the training ground and not what we think they can do.
"We do that all the time, to not go off script.
"It was three of our most experienced boys that end up in that mix up. My words at half-time, calmly, would've been 'when did we work on that one? Because we haven't'.
"But they thought we had an advantage with a three-v-two over there, so I get why they've done it.
"You can't always chastise them, on another day that opportunity doesn't end up in the back of our net. Thankfully today it hasn't burned our lads.
"And later on, a few minutes after, they've atoned for their mistakes. They're honest boys and made a genuine mistake really."
Town's two wing-backs and midfielder Josh Vela - who went off injured with a knock that will be assessed - were involved in the early set-piece breakdown but recovered in otherwise excellent performances.
Cotterill added on the duo: "If you look at the two lads we've got in the wide areas, they are tremendous crosses of the ball.
"They will provide assists and goals throughout the season. I'm not surprised that was how Luke would score his goal and I'm not surprised Elliott would cross it.
"A big well done to those boys, because a few minutes earlier they'd made a bit of a pig's ear of a free-kick, but all is forgiven now."
Striker Udoh replaced the quiet Sam Cosgrove and put in-form Wimbledon to the sword with a fine header midway through the second half, his first goal since a couple of the EFL Cup against Lincoln last month.
Ex-AFC Telford hitman Udoh is a big favourite at the Meadow and it was his name being rung around before and after his match-winner, as Town fans serenaded the players for a well-earned victory at full-time.
"It's a big moment for Dan. Everyone knows what I think of Dan Udoh. But you can't pick a team based on who you like, because if I did I'd pick Dan Udoh every week," Cotterill said.
"Dan had gone off the boil a little bit, also I thought there was added pressure because I think we'd played six games in the league and he hadn't score. It was time to mix things up and change it around.
"Sam came in and scored a couple, Ryan Bowman's come in and scored. That competition is what we need."
One chance for top scorer substitute Jack Rudoni aside, Town saw out the final 20 minutes in impressive manner, defending solidly and keeping the Dons at bay reasonably comfortably.
The boss, for whom the win was his side's second in nine league games this season, said: "It was important to back it (draws) up with a win. For the players, their confidence, and also for the crowd as well. They'll be disappointed we've gone 1-0 down from our free-kick, no-one's more disappointed than me at that.
"But you have to recover and show good steel, mettle and team spirit, because if you don't then you won't get that equaliser and you might end up getting beat."