Edin Dzeko's winning goal came moments after Carlos Tevez fired against the crossbar and James Milner struck the post soon afterwards.
Tevez and Milner also narrowly missed the target in the first half while Jack Rodwell was unfortunate not to convert a number of opportunities.
But West Brom made a positive contribution to the contest and had a number of chances themselves, going closest when Markus Rosenberg rattled the crossbar from a free-kick.
Clarke said: "We didn't have too much time to prepare for the game but the way the players applied themselves was excellent.
"We had chances, Joe Hart made his usual number of good saves.
"Markus hit a free-kick against the crossbar and we were a bit wasteful at times.
"But when you come to these places you have to be on top of your game all the time."