The 26-year-old, making his first league start of the season, kick-started the game, after the two sides had cancelled out each other in a lack-lustre opening half.
Ched Evans equalised after Bednar's opener before the big striker found the target again with Cotterill stepping off the bench to level matters in a nail-biting finale.
West Brom survived an early scare, Darius Henderson reaching Keith Treacy's deep cross as Scott Carson snatched the ball on the line
before centre-back Matt Kilgallon saw his close-range shot blocked by Shelton Martis.
The Hawthorns side, struggling to break down United's tough-tackling defence, escaped again after Treacy was given too much space on the left, the on-loan Blackburn winger drilling in an angled shot which flew inches over the top.
West Brom finally carved out an opening on 32 minutes, midfielder Jerome Thomas linking with Shaun Cummings before firing in a right-foot strike which took a deflection as Mark Bunn dived to his left to cut out the danger.
Little winger Jamie Ward saw his fierce long-rang strike deflected by Swedish centre back Jonas Olsson before £3million striker Evans almost got the breakthrough with a 25-yard strike that flashed a fraction wide of the target.
West Brom took the lead on 54 minutes Bednar playing the ball out to Cummings before racing back into the goalmouth and stretching to meet the midfielder's precise cross beating Bunn from eight yards.
The Sheffield side were level two minutes later when Ward's centre was met by Evans who finished with a close-range header, his second of the season.
Bednar struck again on the hour galloping on after Carson's long kick upfield. He shrugged off a weak challenge from full-back Andy Taylor before lashing a low shot into the far corner from 20 yards, beyond Bunn's despairing reach.
Ward had the ball in the net five minutes later only to be ruled offside before Treacy cut inside from the right flank and fired in an angled shot that was saved by Carson's outstretched feet with Cotterill having the final word.