But what the manager was first to admit was that the first goal was firmly in their control - and it shouldn't have happened.
The game was only three minutes old when ex-Burnley winger Pugh, who dominated the game with his skills and trickery, latched onto a wayward pass and charged down the left of the area.
He played in a perfect cross which Kelvin Langmead just managed to get a touch on but it fell into keeper Kenny Arthur's arms from close range.
However, the usually reliable keeper couldn't keep hold of the ball and spilled it at the feet of West Brom loan striker Stuart Nicholson who had the easy task of firing home from two yards out.
"We should have got rid of the ball initially," slammed Coleman. "It was a decent cross and, give their lad his due, he is a poacher and he scored the goal. After that we were always up against it."
Coleman had stuck with the same side which had beat Grimsby 4-1 the week before but, from the shocking start, they were always chasing the game.
However, it could have been a different story if one of the chances in the first half had gone in.
Speedy striker Roscoe Dsane put in a perfect cross which fizzed across the area but no one was there to touch the ball home. Then Graham Branch's long ball was headed back across goal by Mullin and defender Mark Roberts towered above everyone but could not steer the ball into the goal.
And a crossfield pass by David Brown found Cavanagh racing free down the right wing and his cross was cleared before Dsane could properly connect.
In between all this, though, home midfielder David Hunt was keeping the Reds under pressure at the back with his long throws and set-pieces causing problems every time.
And former Red Darran Kempson was brought on early doors for the ill Colin Murdock and to keep check on the lively Dsane who was proving a handful for the Shrews backline.
But Stanley continued their hunt for the equaliser and Coleman thought it had come when, on the half hour, Harris broke from a Shrews corner, charged down the field and found unmarked John Miles in the area. The winger twisted and turned to make space for himself but agonisingly ballooned the ball over.
And they were almost made to pay immediately when the home side should have made it two on 34 minutes.
Skipper Cavanagh was caught in two minds whether to play the ball back to Arthur with Dave Hibbert lurking. He eventually decided to but the striker nipped in, got the ball and fired past Arthur. However, to spare the captain's blushes, the ball hit the far post and rebounded into the safety of the relieved keeper's arms.
It was an entertaining game on a hot day with barely a minute passing without incident and, on the stroke of half-time, defender Roberts had another great chance to get Stanley back in it when he again jumped the highest to connect with a Miles ball in but again he couldn't find the target.
Then the game changed with Harris' dismissal three minutes after the restart when he hacked down Pugh.
"The directives are clear that if you lunge at a player you run the risk of being sent off but I saw four or five lunges on Saturday and I only saw one red card," said Coleman.
And the ten-men Reds were still reeling when the lively winger was again at the centre of things four minutes later after Cavanagh stuck his foot out and tripped Pugh just inside the area.
Hibbert had no problems despatching the penalty and sending Arthur the wrong way for his fifth goal of the campaign.
Bell was then sent to the stands by the referee and Coleman put Rommy Boco and Shaun Whalley on as they looked to make a difference.
But it was Brown who almost got the Reds back in it with what would have been a 25-yard wonder goal which whistled just over the bar.
Shrewsbury manager Gary Peters then changed his side around and hitman Michael Symes came off the bench and he almost grabbed a third but Arthur came out to block his goalbound effort.
The ten-men though kept going and the ball just didn't fall for them when Wales keeper Glynn Garner dropped a high ball three minutes from time and both Mullin and Boco had a go with the goal empty but the home side managed to block the efforts and keep Stanley out.
"I cannot beat the lads up about their performance," said Coleman. "Apart from the first goal, we actually played quite well and had a lot of the possession with Robbo unlucky with a couple of chances.
"If one of them goes in, it will be a different game but playing on a hot day with ten men is bound to have an effect.
"The lads have to be given some credit for not letting their heads go down. We have got to learn from our mistakes and take the positives from the game."