Boro looked as if they would fall behind in the play-off race after 18 minutes when Wes Thomas drove home for the Cherries and then Scott Malone followed suit.
However, Boro made up for their poor start in the seven minutes prior to half-time as first Freeman rifled in from range and then Bostwick levelled with a superb half-volley.
Bournemouth, who parted company with boss Lee Bradbury on Sunday, got off to a perfect start under caretaker manager Paul Groves as Thomas picked up from Harry Arter and slotted home.
Stevenage struggled to get a grip on the game and 18 minutes later full-back Malone doubled the Cherries' advantage with an effort from all of 25 yards. But their two-goal cushion lasted just seconds as Freeman rifled a fierce long-range drive high into Darryl Flahavan's goal before Bostwick beat the 'keeper with a brilliant half-volley.
Midfielder Lawrie Wilson went close after the restart as Stevenage pushed for a winner while Thomas squandered a gilt-edged chance to steal it for Bournemouth.
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