The Baggies trailed their visitors 2-0 at half-time after goals from Christian Benteke and Gabriel Agbonlahor, but a marked improvement in the second period saw Clarke's men earn a point through Chris Brunt and Peter Odemwingie.
Clarke said afterwards: "I'm not sure relief was the emotion. The emotion was disappointing at the way we played in the first half.
"We knew Villa are a dangerous team but we didn't show our qualities in the first half. We didn't pass the ball like we can and it was nice to get in at half-time and regroup.
"The second half was much improved and there was a little bit of pride at the way we played."
On rumours Odemwingie will leave the club in the January transfer window, Clarke added: "No-one is going out in January. We don't need to sell anyone.
"Will he move if there is a silly offer? You don't know our chairman (Jeremy Peace). No-one is going out. The quicker we get to February 1 the better. No-one is coming in and no-one is going out."