Dwight Gayle struck the game's crucial goal midway through the first half.
The striker capitalised on a glaring error from Goran Popov as his hacked clearance fell straight to Hiram Boateng, who found Gayle and he simply wrong-footed Ben Foster with a left-footed strike from the edge of the penalty area.
Substitute Marouane Chamakh converted a second goal with the last kick of the game deep into injury time, firing into an empty net.
Barry Bannan and Craig Dawson both hit the woodwork in an evenly-matched first half, before Palace goalkeeper Julian Speroni turned in a man of the match performance after the break to deny dominant Albion.
A frustrated Downing said: "At half-time we talked about a lack of intensity to our game, we were a bit disappointing without the ball, we didn't get after people.
"We'd been positive in the last four games and I just felt there was too much caution here, waiting for things to happen.
"We grabbed the game to a certain extent in the second half but with just not enough creativity in the final third."
A crowd of just 12,700 watched the third round tie. Asked whether Albion's lack of a spark was down to the lack of atmosphere, Downing replied "No, I don't think that's a reason or excuse.
"We have to generate our own atmosphere, momentum and pressure.
"I don't think we did enough of that, particularly in the first half."