Albion's win in Saturday's late kick-off leaves Newcastle needing a point in Monday's clash with Plymouth to secure the Championship trophy.
The Baggies went into the game knowing they needed a victory to keep their slim title hopes alive and they grabbed the three points courtesy of strikes from Cox and Bednar on 45 and 55 respectively.
Despite already sealing promotion, Albion pushed forward from the start and could have taken an early lead through Cox but his 20-yard strike was well saved by Brad Jones.
The Australian stopper was called into action once again moments later, saving Bednar's deflected effort. Boro tried to counter West Brom's early pressure but their assault continued as Chris Brunt forced yet another save from Jones.
The North-East club's defensive woes remained and they nearly paid the price in the 29th minute when Robert Koren robbed on-loan Tottenham defender Kyle Naughton before playing in Bednar who rattled the crossbar after cleverly working an opening.
Gordon Strachan's counter-attacking was almost rewarded minutes later, though, as they turned defence into attack allowing Barry Robson to race through only to see his shot blocked by Gabriel Tamas.
The visitors again threatened just before the interval as Chris Killen cleverly found Scott McDonald who looped an effort just over Scott Carson's crossbar.
Despite the warnings, Roberto Di Matteo persisted with their attacking approach and they took the lead on the stroke of half-time through Cox's 10th goal of the season. A hopeful long ball found Bednar who chested into the path of Cox, allowing the former Swindon man to jink past Stephen McManus and fire an emphatic strike into the top-left corner.
Albion continued where they left off at the start of the second half as Cox saw his effort blocked after an excellent one-two with Bednar.
They doubled their advantage in the 55th minute when Cox nodded Koren's cross into the path of Bednar who made no mistake from five yards.
As the game wore on, Middlesbrough's players must have feared a repeat of the 5-0 drubbing they suffered in the reverse fixture as the Baggies continued to dominate.
The Black Country club nearly made it three midway through the half but Jones was again on hand to deny Koren. The 29-year-old Slovenia international turned provider moments later, as his clever through-ball found Bednar who saw his effort tipped over the bar by the impressive Jones.
After another couple of fine saves from the 28-year-old, Boro could have given themselves a sniff in the 85th minute but Leroy Lita could not get the ball under control allowing Carson to collect.
Their efforts were thwarted once more five minutes later when McDonald headed Robson's free-kick against the post. That proved to be the game's last opportunity as the hosts saw the game out.