© AP Photos
Daniel Agger then had a fine chance to break the deadlock, but he failed to connect with Downing's low cross inside the box.
Steven Gerrard was the next to have a chance for the Reds when Downing poked a ball through to the skipper, but he was denied by a sprawling Foster save from six yards out.
Foster was called into action again three minutes later, having to get down well to his right to deny a clever flick from Jordan Henderson.
The Baggies keeper continued his fine form with two saves in the space of three minutes. First, he was at full stretch to deny Fabio Borini's effort, holding on to the ball for good measure.
© PA Photos
He then pulled off an even better save to deny Gerrard's powerful strike from the edge of the box.
Liverpool were handed the perfect chance to finally take the lead with just over 10 minutes of the match left when Jonas Olsson was adjudged to have pushed Luis Suarez inside the box. Gerrard stepped up for the spot kick, but Foster again frustrated the Liverpool captain by saving low to his left.
West Brom had their first shot of the match in the 80th minute when Youssouf Mulumbu was denied by a flying Jose Reina save, but the Baggies took a shock lead from the resulting corner, Gareth McAuley heading Chris Brunt's cross in off the bar.
The visitors then completed their smash and grab win by doubling their lead in stoppage time, Romelu Lukaku beating Agger before slotting the ball past Reina.