The Cherries striker, who now has 19 goals for the season, also missed three easy chances as the visitors surrendered after a promising first half.
Pitman twice failed to score in the opening 45 minutes after being played clear of a static Barnet defence, and then immediately after the break shot over the bar from the edge of the area, again with only goalkeeper Jake Cole to beat.
Bournemouth had to wait until the 59th minute before making the breakthrough when Liam Feeney crossed from the right and veteran Steve Fletcher headed home from six yards.
Two minutes later Cole and defender Joe Devera got in a tangle on the edge of the area to let Pitman in for his first.
Pitman then grabbed his second of the night in the 76th minute, drilling home a right-footed shot from 15 yards as dispirited Barnet - who lost John O'Flynn to injury following a collision just after the hour mark - failed to muster a single shot on target.