Bournemouth dominated the first half and deserved their lead in the 38th minute when Danny Thomas seized onto a mistake by Tim Ryan to score with a low shot.
That dismissal instantly put a score of 300-plus out of range, though Tim Bresnan (28 from 27 balls) did his bit, while Shahzad and Tremlett each hit a six.
Barritt's try will surely be a contender for score of the season after a flowing move from inside the Saracens half but Sharples quickly hit back after Tim Molenaar blasted through the visiting defence.