Johnson was shown the yellow card twice in a three-minute spell - first for fouling Mark Viduka and then for vigorously protesting that Lucas Radebe had handled.
Albion's penalty claims were rejected by referee Uriah Rennie, and he had no hesitation in ordering Johnson off.
Robbie Fowler, who this week rejected a move to Manchester City, was given just 11 minutes action by Leeds manager Terry Venables who sent him on for Paul Okon too late in the game to have much effect.
With Jonathan Woodgate injured, Radebe made his first start for Leeds since mid-November and Teddy Lucic also returned to the defence in place of Danny Mills, who was on the bench.
Albion, who made four changes, were without Adam Chambers, Larus Sigurdsson, Ronnie Wallwork and the suspended Jason Roberts. In came Scott Dobie, Derek McInnes, Igor Balis and Sean Gregan.
Dobie should have given Gary Megson's team a 15th minute lead when Neil Clement's free-kick rebounded to him off Darren Moore, but the Scotsman shot wide from 12 yards.
Dominic Matteo marked his 100th appearance for Leeds with an accomplished performance alongside Radebe, but Albion's ultra-defensive tactics helped to spoil the game as a spectacle.
Twice in the first half Viduka was denied by Russell Hoult's agility and Alan Smith had a goal-bound volley superbly saved, but with West Brom packing men behind the ball, Leeds had to work extra hard to create openings.
The visitors lost Balis in the 35th minute to be replaced by Wallwork who immediately created a chance which both Dobie and Daniele Dichio spurned.
Leeds replaced Eirik Bakke and Jason Wilcox with Seth Johnson and James Milner in the 69th minute but they couldn't force a breakthrough.