Luke Moore put Swansea in front only for Romelu Lukaku to equalise with his 13th goal of the campaign before an own goal from Jonathan de Guzman earned Albion victory.
But Swansea boss Michael Laudrup slammed referee Lee Mason after he disallowed a last-gasp effort from Roland Lamah.
Replays showed the ball struck Baggies goalkeeper Ben Foster and defender Gareth McAuley before substitute Lamah put the ball in the net.
But Mason ruled Lamah was offside and West Brom hung on for their third win in four Barclays Premier League matches.
Clarke felt the goal should have been given and said: "We got a lucky break. It should have been given. It was a goal. I think it came off Ben Foster last so that is fortuitous."
However, Clarke felt Albion deserved the victory which kept alive their hopes of securing a European spot.
He added: "We got a lucky break but sometimes when you put in the effort, energy and quality we did into that game...I think we earned that little break.
"We deserved to win the game. Don't be writing down that we didn't deserve to win the game.
"I'm delighted because we came from behind. We don't do it very often and in the Premier League there are not too many teams that can do it. When you go behind in this league, it is difficult - especially against a team with the quality of Swansea."
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- Roland Lamah