Town saw an early lead earned via Luke Leahy's header cancelled out by Aiden O'Brien's close-range flick on 29 minutes, just seconds before midfielder Vela was shown a straight red card for a late swipe at Ronan Curtis.
The hosts had already lost fellow midfield man David Davis to injury at Montgomery Waters Meadow. Davis limped off with what looked like a painful ankle problem and Vela will be sidelined for three matches after Cotterill confirmed Town will not appeal his suspension.
Shrewsbury looked good for a well-earned point at Montgomery Waters Meadow as Danny Cowley's visitors struggled to make their extra man advantage count until former Town loanee Louis Thompson, who spent a brief period at the Meadow from Norwich in 2019/20, curled in a fine late finish over Marko Marosi from outside the penalty box.
The home defeat means Salop are now winless in eight league games - nine in all competitions - and remain 18th in League One, four points clear of safety.
The Town boss felt Vela, who received Town's sixth red card of the campaign, had been 'tremendous' until his dismissal and was looking to impress and make up for his part in Plymouth's winner in Shrewsbury's last fixture, the 1-0 Home Park defeat 10 days earlier.
Cotterill said: "It's a tough one tonight, a tough one. I thought they were brilliant tonight, the boys.
"With 11 men I thought we were excellent, started off really well and played some great football, created quite a few chances in that first 30 minutes.
"We need to probably get a second one but we're a work in progress with that.
"It was a great start and (if not for the) injury to David Davis we think we're riding that one out.
"Tyrese came in and has done OK in there.
"Then the sending off, I don't think I have any complaints with the sending off. It wasn't the high tackle, I think it was just the speed that Josh has gone in. It wasn't a high tackle.
"I don't think that one is appealable, I think we have to take that one on the chin.
"And by the way, before he got sent off, I thought he was absolutely tremendous tonight. I think he was trying to make up for maybe not marking that man on the edge of the box against Plymouth.
"I think he wanted to do really well. His all-round game up until then just led to a little bit of over-exuberance and it took him into a challenge that resulted in him getting a red card.
"From then on, after 30 minutes, we know we've done it before but we're not always going to come up with a victory or a draw when you've done it before, there's no guarantees.
"Having said that, when we got to the stage we'd got to, I thought we'd seen off what they had to throw, but unfortunately not.
"The sub did two chops on the edge of the box that worked out boys' thighs, when they're tired, and we couldn't get out to get a block on the shot, which if it's a mistake is probably the only mistake we made once we had 10 men.
"So it was really tough tonight, and the supporters I thought were brilliant, I thought they backed the boys all the way which is really important, so I feel the players and fans tonight."
There was some controversy with Pompey's equaliser through O'Brien, who flicked in a Denver Hume strike after Marosi's punch did not fully clear Town lines.
Defender Ethan Ebanks-Landell scrambled to clear off the line but referee Sebastian Stockbridge's assistant waved his flag immediately to suggest the ball had crossed the line.
"You can't open up (with 10 men), then you can get beat four or five, they have very good individuals. I certainly felt with 11 men on the pitch we were the better team and would've gone on to win the game," the manager added.
"You can always pick faults with any goal you concede, I'm not going to hang anybody with any goals we've conceded. But it looked like the ref and linesman couldn't wait to give it, I felt, because it didn't look that clearly over the line to me."
Cotterill had no further diagnosis on what appeared a painful ankle problem for Davis. The experienced defensive midfielder could put no weight on his right foot as he was helped off in the first half.