The Baggies looked to be cruising to victory when they went 2-0 up within nine minutes of the kick-off through goals from Morgan Amalfitano and Dorrans.
However, the Bluebirds dragged themselves back into the game on the half-hour through Jordon Mutch's strike, and levelled 17 minutes from time when defender Steven Caulker powered home a header.
But there was still drama to come when Thievy Bifouma looked to have snatched victory for the home side three minutes into added time, only for Mats Moller Daehli to deny them with virtually the last kick of the game.
Afterwards, Scotland international Dorrans told Sky Sports: "It's very disappointing. It feels like a defeat - everyone is very unhappy because it would have been a massive three points for us.
"Sadly, it wasn't to be and we conceded a silly goal near the end but we have to stick together and move on to the next game (at Norwich). Hopefully we can pick up points then.
"The first half was probably the best we've played all season - we could have been three, four or even five up. It wasn't to be and we move on."