West Brom will play in the Premier League next season, where they could yet be joined by Swansea City, but Wigan Athletic are in League One.
With Brentford having faltered earlier, Fulham beat Sheffield Wednesday to keep their top-two hopes alive.
Hull City shipped eight goals to Wigan Athletic, whose survival hopes look increasingly positive at a time of worrying upheaval at the club.
Leeds United were displaced at the top of the Championship by West Brom, who defeated Derby County 2-0 on Wednesday.
Hull City fought hard but were beaten at West Brom, who move back to within a point of Championship leaders Leeds United.
Leeds United's advantage at the top of the Championship was cut to one point as West Brom claimed a 3-0 victory away at Sheffield Wednesday.
West Brom dominated against Birmingham but could not score, while Louie Sibley and Rhian Brewster produced heroics for Derby and Swansea.
Sheffield Wednesday failed to muster a single shot on target in their 1-0 defeat to Manchester City, much to Pep Guardiola's delight.
SHEFFIELD WEDNESDAY 0 MANCHESTER CITY 1 Pep Guardiola gembira juara bertahan Piala FA, Manchester City, tiada masalah untuk meraih pusingan suku akhir dengan kemenangan 1-0 ke atas Sheffield Wednesday.
Holders Manchester City booked their place in the quarter-finals of the FA Cup with a 1-0 victory over Sheffield Wednesday at Hillsborough.