The teenager fired in a left-footed volley in the 88th minute after his initial drive was blocked to condemn Bournemouth to their fifth straight defeat.
In a hard-fought contest, it was the Cherries who started the brighter with skipper Steve Cook rattling the crossbar with a fifth-minute header.
Veteran Neil Sullivan, 43, making his first appearance for Doncaster this season, was the busier goalkeeper throughout, producing a number of smart saves from Brett Pitman and Lewis Grabban.
Former Leeds striker Billy Paynter scored his 11th goal of the campaign to open the scoring for Rovers after converting a pin-point David Cotterill cross after 66 minutes.
Bournemouth responded and thought they had picked up a vital point in the promotion run-in when Matt Tubbs lobbed over a stranded Sullivan in the 85 minute.
And after Husband's late winner, Eddie Howe's men were centimetres away from grabbing a last-minute equaliser when Brett Pitman's header cannoned back off the post in stoppage time.
