The Birmingham goalkeeper pulled off a series of fine saves and denied substitute Atdhe Nuhiu in stoppage time before blocking the striker’s follow-up effort.
Wednesday had dominated the first period, during which Steven Fletcher stretched to curl the ball past Camp for the only time the Birmingham keeper was beaten.
Che Adams equalised early in the second half before Camp produced his one-man show to earn his side an away point.
The home side carved out the first real opening but when Barry Bannan floated the ball to the far post, Adam Reach - who had recovered from a hamstring strain to start - volleyed well off target after eight minutes.
Marco Matias beat the offside trap from George Boyd’s ball over the top but Camp raced from his area to clear as Wednesday continued their fast start.
Bannan then swung in a corner from the right but Michael Hector could only steer the ball wide.
The home side deservedly took the lead in the 18th minute as Fletcher curled his foot around the ball to beat Camp after Bannan’s free-kick had been headed back across the area by Tom Lees.
The closest Birmingham came was when Wednesday cleared a corner but Maikel Kieftenbeld volleyed off target just before the half hour mark.
Another Bannan ball sent Matias clear but when he shot from a narrow angle Camp beat the effort away for a corner with 10 minutes of the half remaining.
Fletcher was penalised for a push on Wes Harding before the forward’s header was cleared off the line.
Lukas Jutkiewicz shot just wide following a quick free-kick routine as the Blues finally threatened the Wednesday goal in first-half stoppage time.
And Birmingham levelled three minutes into the second half when Jutkiewicz headed on a long ball and Adams beat his marker to the ball before curling past Keiren Westwood for his 12th goal of the season.
Camp kept the scores level when he dived to his left to keep out a smart shot from Reach on the hour mark.
Next Camp pushed away a Fletcher header and Reach fired the rebound wide, while at the other end Jota shot wide when in a good position as a clever build-up came to nothing.
Camp blocked another Fletcher shot from a narrow angle with his leg as the Birmingham keeper again kept his side in the game.
Jutkiewicz headed wide from a flighted ball in by substitute Craig Gardner during a rare Birmingham attack.
And the Blues could have taken the lead with under three minutes remaining but Gardner’s low free-kick hit a post with Westwood well beaten.
But there was still time for some late Camp heroics as he twice denied substitute Nuhiu in stoppage time.
Sheffield Wednesday: Westwood, Palmer, Lees, Hector, Fox, Hutchinson, Boyd (Winnall 89), Bannan, Reach, Matias, Fletcher (Nuhiu 89). Unused substitutes: Pelupessy, Baker, Dawson, Pudil, Penney.
Birmingham City: Camp, Harding, Morrison, Dean, Colin, Jota, G Gardner (C Gardner 63), Kieftenbeld, Maghoma (Mahoney 46), Jutkiewicz, Adams. Unused substitutes: Scarr, Dacres-Cogley, Trueman, Lubula, Lakin.
Referee: Andy Woolmer (Northamptonshire).