Town played more than half after a straight red for David Davis in first-half stoppage time, after already trailing through Alex Pritchard's opener.
But threadbare Shrewsbury, with five players aged 18 or under in their squad, made light of their low numbers and man disadvantage and claimed an impressive point through Udoh's rasping half-volley with 25 minutes to go, his fifth goal of the season.
Town fans already knew their side would be without the suspended Elliott Bennett and injured duo Shaun Whalley and Josh Daniels.
But the stark nature of Shrewsbury's availability woes were laid bare as centre-half pair Ethan Ebanks-Landell and Aaron Pierre were also ruled out of the clash through injury.
The hosts retained their three-man defence, with skipper for the night Luke Leahy surprisingly playing in the middle of the trio.
Rekeil Pyke was handed a rare league start and operated out of position at right wing-back. Teenager Tom Bloxham supported the front two in attack.
Even more startling was the home bench, which featured first-year scholars Ben Kaninda and Kade Craig, both defenders, who had never previously featured in a senior squad. They were two of four outfield players aged 18 or under on the bench.
Sunderland appeared to have stemmed their run of poor form with a welcome 2-0 win over fellow big-hitters Ipswich last time out.
It was a lively, open start on a bitter night in Shropshire in front of an atmosphere helped by three quarters-full away end.
Threadbare Shrewsbury signalled their intentions early on that they were not just to sit back and be attacked.
The lively Nathanael Ogbeta flashed a half-volley wide of former Bayern Munich goalkeeper Thorben Hoffmann's right post.
And it was the hosts inches away from a 10-minute lead as Ryan Bowman's low strike deflected inches wide of the post with the keeper stranded after Daniel Udoh had pressed back possession.
But by 16 minutes the visitors had the lead and it came through the classy right foot of Pritchard. He was allowed to venture into the box from the left and given too much space to bend an excellent finish above Marko Marosi and into the far corner.
Prospects of a rare clean sheet had gone but Town responded well. Ogbeta was a regular attacking outlet down the left, but end product left something to be desired.
Bloxham looked lively attacking down the right, while Udoh was full of beans for Town with skill and purposes in his play.
Marosi was at his best to fingertip Lynden Gooch's strike from distance just wide on half hour before Ross Stewart headed over from a corner.
Town gave as much as they got in a watchable contest, but things bubbled over before the break.
Nathan Broadhead was fortunate to escape a red card for kicking out at Luke Leahy off the ball just before stoppage time. He was cautioned.
Before, in time added on, the red card was issued for Town's David Davis. The experienced midfield anchorman steamed into a challenge on Broadhead with at least one foot off the ground. Sunderland players were immediately unhappy and referee Peter Wright reached for his red.
Incessant boos greeted the officials at the break, while Cotterill and Aaron Wilbraham waited for a response.
Town started the second half well as Udoh's shot deflected wide after Bloxham thumped over from distance.
But Sunderland's man advantage soon told as first Dan Neil shot wide from 25 yards before Stewart headed wide, Pritchard shot across goal and Luke O'Nien nodded off target.
Shrewsbury looked pinned back but Bowman was inches from connecting to Ogbeta's deflected cross for a leveller on the hour.
And, just four minutes later, Udoh lit up the Meadow.
Sunderland were unable to clear a bouncing ball in the box and the popular frontman took matters into his own hands. With little backlift he rifled in a crisp half-volley from the edge of the box before bedlam ensued behind the goal.
The equaliser stunned the visitors, who had been on top but despite dominating possession afterwards created very little. Marosi made a strong one-handed save to deny O'Nien, but hard-working Town were resolute.
Broadhead's volley crashed the near post from an impossible angle but the hosts impressed in keeping their visitors at arm's length for a valuable point.
Shrewsbury Town (3-4-3):
Marosi; Pennington, Leahy, Nurse, Pyke, Vela, Davis, Ogbeta; Bloxham (Cosgrove, 83), Udoh, Bowman.
Subs not used: Burgoyne, Kaninda, Craig, Wilson, Leshabela, Caton.
Hoffmann; Winchester, Doyle, Wright (c), O'Nien; Gooch (McGeady, 75), Neil; O'Brien (Evans, 62), Pritchard (Dajaku, 75), Broadhead; Stewart.
Subs not used: Patterson, Flanagan, Alv, Embleton.
Attendance: 6,253 (838 Sunderland fans)
Referee: Peter Wright