Mikel Arteta converted both opportunities from 12 yards in either half but Clarke was left distinctly unimpressed by the performance of the match official, particularly in relation to the first spot-kick.
Gunners midfielder Santi Cazorla fell to the floor over Steven Reid's out-stretched leg in the 26th minute, with Clarke insisting there was "clear daylight" between the players.
And while the Baggies boss accepted that Alex Oxlade-Chamberlain was felled by Chris Brunt for the second goal in the 64th minute, he was adamant that the visitors should have been awarded a free-kick of their own moments earlier.
Clarke told Sky Sports: "I'm not angry, I'm frustrated. The first penalty decision, for the life of me, I can't understand why the referee would give that. He has to be sure there is contact.
"There is clear daylight between Steven Reid and Cazorla, he goes down, and the referee has given him what he wants.
"The referee today wasn't good to us. I think there was a penalty for us in the first half - a handball from (Per) Mertesacker - but the officials took the easy way out and gave the free-kick to Arsenal for no reason.
"Even on the second penalty - which I've got no complaints about as the tackle from Chris Brunt on Alex Oxlade-Chamberlain was a foul - but before that Goran Popov was fouled in the build-up.
"I think the referee has to be in a position where he can judge. I knew when Cazorla went down that there was no contact, you could see. The way he went down, people don't fall like that.
"It is down to the referee to make the right call. Unfortunately for us today he didn't make the right call on that one and he didn't make the right call on a number of other ones."