Everton's only serious response was a header from midfield man Jack Rodwell that flew just wide and Moyes made an alteration the situation demanded at the interval, removing defender Tony Hibbert in favour of striker Jermaine Beckford.
As half-time approached, Jim O'Brien had a good effort on goal from deep inside the penalty area that drifted wide, but it was Falkirk who enjoyed the lead at the interval.
Morne Steyn finally levelled the scores on 36 minutes following an England infringement at the ruck, but the fly-half spurned the opportunity to take a three-point lead at the interval when he uncharacteristically dragged a 25m drop-kick attempt wide of the posts.