The Gunners were leading 2-0 through Tomas Rosicky's brace when Per Mertesacker was sent off for a professional foul on Shane Long inside the box.
James Morrison despatched the resultant spot-kick but West Brom could not find a leveller.
Clarke said: "I thought we played well. They scored in the first half with one of the very few chances they created. We had good control in the first half but didn't really create enough.
"We could have had a penalty for a challenge on Claudio Yacob on the edge of the box (by Laurent Koscielny). On another day that was a penalty and would have changed the complexion of the afternoon.
"We were disappointed to lose the second goal at the start of the second half. It was crucial for us to get back in it and we gave away another soft goal, a bad goal to lose.
"At 2-0 it is difficult to chase the game. We made a couple of changes. Then the penalty and the red card does change the game and gives us the chance to get something from the game."
Clarke insists eighth-placed Albion will not allow their season to end tamely.
He said: "If you had asked me that after the West Ham game last week, I would have said yes.
"The response was good. The performance deserved to get something from the game. I don't think you can say we took it easy or didn't display a determination to get something from the game."