Cox scored his tenth goal of the season in first-half stoppage time after being set-up by Bednar and he then repaid the favour by creating the second for Bednar to secure a victory that delayed Newcastle United being crowned as the Championship title winners.
But the damage was rather more terminal for Middlesbrough who also saw their eight match unbeaten run come to an end.
They will now face a second successive season outside the top-flight as they are seven points behind sixth placed Swansea with just two games remaining.
Middlesbrough had received a pre-match boost when they agreed a dealwith Portsmouth that enabled them to include captain Gary O'Neil in their side.
Middlesbrough had been due to pay Portsmouth £1m when O'Neil made his 100th appearance for the club.
He had been left out of the previous two games to avoid reaching that milestone but faced Albion after Middlesbrough agreed a compromise deal for the remaining three matches of the season.
But the presence of their skipper didn't do much to inspire Gordon Strachan's side who had been beaten 5-0 by Albion at the Riverside Stadium last September.
Albion dominated affairs and, but for goalkeeper Brad Jones and the woodwork, they would have secured a more resounding victory.
Jones set the tone as early as the second minute when he dived bravely at the feet of Cox while Rhys Williams briefly raised Middlesbrough spirits when he shot over when well-placed but it was Albion who carried by far the greater threat.
Jones beat away a 19th minute drive from Cox before Bednar had a shot blocked two minutes later.
Chris Brunt was then denied by Jones before Bednar hit the cross bar after Robert Koren had hustled the ball away from Kyle Naughton.
Middlesbrough seemed content to try and hit Albion on the break and that tactic almost paid off five minutes before half-time when Chris Killen picked out Scott McDonald only for the Australian international to fire wide from the edge of the area.
Having survived that let-off, Albion finally made their pressure pay when Bednar collected a long pass from Gabriel Tamas.
Bednar teed up Cox on the edge of the area and the former Swindon striker beat Jonathan Grounds and Stephen McManus before firing a rising right-foot drive beyond Jones.
Cox turned provider 10 minutes into the second-half after McManus had failed to fully clear a Koren cross.
Koren's second centre was nodded down by Cox into the path of Bednar who rattled home his 11th goal of the season from close-range.
Jones then pulled off two more smart saves from Bednar and Koren to prevent Albion racing to the margin of victory that their domination had threatened before
Middlesbrough's dismal day was summed up in added time when McDonald hit the post with a header from eight-yards.