Despite dominating the game, Sir Bobby Robson's much-changed side only managed to beat the outstanding Russell Hoult on one occasion and paid the penalty.
A Shola Ameobi own goal gifted the Baggies a first-half lead that was eventually cancelled out by a stunning 65th minute equaliser from Laurent Robert.
But in extra-time a wayward Andy Griffin pass found United on the back foot and Lee Hughes' clinical finish earned the Hawthorns outfit a money-spinning fourth round tie against last season's beaten finalists Manchester United.
But for the heroics of Hoult, Newcastle could have won easily as they piled up a staggering 23 corners throughout the 120 minutes.
With the inspirational Lomana LuaLua causing all sorts of early problems for the West Brom defence a United goal looked inevitable.
Just 56 seconds were on the clock when Hoult beat away a Robert thunderbolt and LuaLua curled another effort marginally wide.
It was an unlucky 13th minute when Steve Caldwell headed Robert's corner on and LuaLua headed against the bar before a magnificent raking pass from Hugo Viana allowed Ameobi a run at the defence, but his effort flashed across the face of the goal.
Sir Bobby was furious when a push in the back of Jermaine Jenas failed to win United a 28th minute penalty and the Baggies raced straight up field to score.
Artim Sakiri's left wing corner was headed on by skipper Sean Gregan and Ameobi missed the ball with his right foot and it cannoned off his left leg and into the net.
Another LuaLua run presented an inch perfect cross for Ameobi after 33 minutes but from just eight yards he headed against a post in a frustrating opening half.
Gary Speed replaced Viana at the break as Newcastle sought an equaliser and a spectacular LuaLua overhead kick flew just over the top.
The overdue equaliser arrived on 65 minutes with Robert working a one-two with Ameobi before drilling in his fourth goal of the season.
Hoult smothered a LuaLua effort before the issue was settled on 101 minutes - 17 minutes after Lee Hughes had replaced Rob Hulse.
Griffin's pass was intercepted and Neil Clement charged forward and played in Lee Hughes to give acting-skipper Steve Harper no chance.
Sir Bobby was disappointed and said: "It was a bizarre night and we missed a lot of good chances while they took their two. I thought our players were terrific and some of our football was tremendous.
"I feel sorry for our players and the fans, and that crowd of 47,000 was unbelievable."