Memorial didn't go away quietly, though, pulling to within two on Obi Emegano's free throws with 1:21 remaining.
Kendall Parks then missed a pair of free throws with a chance to stretch the lead to two possessions.
But Lee Cooper missed a driving layup attempt, and McClish snagged the rebound to keep the Bombers up.
McClish missed the first free throw but made the second to put Midwest City up by three, 50-47.
Memorial looked to be holding for a final shot in the closing seconds when Woodard pulled up well behind the line and drilled a 3-pointer with three seconds left to tie the game for the first time since the very early stages.
McClish then just missed a half-court attempt to end the game.
The win was Midwest City's third over the Bulldogs this year.
The Bombers have not played Putnam West this season other than a preseason scrimmage.
Midwest City entered the game as the hottest team in Class 6A after beating No. 1 Edmond Santa Fe in the area championship game last week.
And the Bombers jumped out early against Memorial, paced by McClish's 10 first-quarter points, to go ahead 17-11 after eight minutes.
High school basketball tournament coverage sponsored by United Supermarkets.