The first dismissals came after only four minutes when West Brom's Andy Johnson and Stoke's John Eustace were jostling for the ball in the middle as Jason Koumas was taking an Albion corner, and the Whitley Bay official opted to give them both their marching orders.
Stoke were reduced to nine men in the 74th minute when defender Marcus Hall was dismissed for chopping down Dobie but that was a decision totally justified by the referee.
Afterwards Albion Boss Gary Megson said: "To send two players off for something like that is beyond belief. I think he made a mistake and I am sure that when the referee views it again he will see what happened in a different light."
He added: "At every corner there are always a lot of players who are challenging in the middle for the ball. To send two men off regardless as to whether it is in the second minute or the 82nd minute for something like that is nonsense."
Stoke manager Tony Pulis said that they would be appealing against the red card for Eustace if they felt it was right to do so after watching the match video.
He said: "I would rather talk about the players and the way we played. The referee did not cost us the game - it was a mistake by one of my men that did."
Pulis added: "I thought we played some good stuff and I am disappointed to have got nothing out of the game."
Dobie secured victory in the 59th minute to end Albion's run of 11 league defeats and six draws against their near neighbours when he found the target from a fine pass from Robert Hulse.
Albion were unlucky in the first half when Koumas rattled the Stoke crossbar with a terrific 25 yarder.
After the break, keeper Ed De Goey did well to block a powerful free kick from Neil Clement.
At the other end Russell Hoult had to stretch to fingertip over the bar a powerful 25 yarder from sub Kris Commons with just seven minutes left on the clock.
West Brom: Hoult, Gregan, Gaardsoe, Gilchrist, Haas, O'Connor, Johnson, Koumas, Clement, Dobie, Hulse. Subs: Murphy, Dichio, Sakiri, Volmer, Berthe.
Stoke: de Goey, Thomas, Williams, Marcus Hall, Russell, Andrews, Henry, Eustace, Clarke, Noel-Williams, Iwelumo. Subs: Hoekstra, Commons, Cutler, Neal, Akinbiyi.
Referee: G Laws (Tyne & Wear)