The former England midfielder diverted Danny Hollands' shot past Nathan Abbey in the 78th minute after Dons midfielder Flavien Belson was earlier sent off.
Jemal Johnson wasted the best chance of a quiet first half when he shot over from close range after Bournemouth's defence failed to clear Dean Lewington's cross.
Immediately after half-time there was controversy as Belson received a straight red card for a late challenge on substitute Brett Pitman, before after 69 minutes Shwan Jalal pulled off a fine save to keep out Kevin Gallen's point blank header.
Bournemouth came close to opening the scoring in the 75th minute when Anderton's 25-yard strike cannoned off the top of the crossbar. But three minutes later the midfielder struck to send Bournemouth through and give the struggling League Two side a much needed boost.