Already-relegated Darlington, on their best run of form all season, put up a formidable defensive barrier for 56 minutes to frustrate Bournemouth.
But then Liam Feeney crossed from the left and Robinson headed home at the far post to set up the win that moves them eight points clear of fourth-placed Rotherham.
Darlington had no answers and conceded a second after 71 minutes. Defender Simon Madden handled under pressure from Feeney and Pitman stepped up to score with a spot-kick for his 26th of the season.
Bournemouth went close to adding further goals with goalkeeper Shane Redmond saving a low shot from Rhoys Wiggins and in the last minute Alan Connell struck the bar.
Darlington had few chances to shine or add to their sequence of 10 points from the last 15. But leading goal scorer Tadgh Purcell lobbed wide from a good position in the sixth minute and then headed onto the roof of the net near the end.