West Bromwich Albion 4 Newcastle United 2
Dorrans struck twice from the penalty spot after Jonas Olsson had given Albion a 17th minute lead.
But Newcastle, who saw Andy Carroll strike twice in the second half, were furious with referee James Linington.
They felt they should have had two penalties of their own in the second period for challenges on Ryan Taylor by Dorrans and Shola Ameobi by Gabriel Tamas.
To add to their anger, when their appeals were rejected after Ameobi went down Albion surged straight up field in the 69th minute and were awarded their second penalty when Taylor pulled down Jerome Thomas.
Taylor's anger got the better off him and he was dismissed for dissent after taking his complaints too far, before Dorrans took his goal tally for the season to ten.
Albion then took advantage of their extra manpower when Robert Koren and Roman Bednar combined to tee up Thomas who pounced at the far post in the 76th minute.
It was a case of third time lucky for Albion who had fought out two Championship draws with Newcastle - the latest of which was only last Monday at St James' Park.
Albion retained the side that started that 2-2 draw while Newcastle made five changes and that ultimately proved to be crucial as United slipped to their first defeat in 15 games.
Albion had chances to take the lead before Olsson broke the deadlock. Tamas was denied by Tim Krul, Koren fired wide from close range before Bednar lifted his shot over the bar.
Newcastle found early chances hard to come by and when Carroll headed wide in the 15th minute following an excellent cross from Jonas Gutierrez, they were punished by Olsson.
The Swedish central defender rose to powerfully head home a 17th minute corner from Chris Brunt.
Jose Enrique did try and clear the ball, but the assistant referee ruled that Olsson's header had crossed the line.
Newcastle's players didn't complain about that decision, but they were fuming 11 minutes later when Dorrans doubled Albion's advantage from the penalty spot.
Hungarian defender Tamas Kadar tangled with Bednar on the left-hand side of the Newcastle area and Linington had no hesitation in pointing to the spot.
Newcastle's players were incensed and delayed the taking of the spot-kick by confronting Linington and his assistant. But Dorrans kept his nerve when the penalty was eventually taken and beat Krul with a powerful drive.
Newcastle sent on Ameobi at the start of the second half and his presence made a difference.
Carroll's fourth goal of the season, when he volleyed home a cross from Gutierrez in the 62nd minute then gave Newcastle hope.
But their comeback hopes disappeared when Dorrans converted his second penalty before Thomas set the seal on the victory.
Carroll then struck injury time strike when he held off a challenge by Youssouf Mulumbu, but it was little consolation for Newcastle and their angry fans.