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Referee Darren Deadman was booed off the pitch at the end after a series of yellow cards to go with Anderton's dismissal.
After a goalless first half, Blackpool burst into life after the break with two goals in the first seven minutes.
Bournemouth midfielder Danny Hollands lost possession in midfield to Adrian Forbes whose smart pass put Keigan Parker in space to score with a low shot.
Blackpool went further ahead in the 52nd minute and this time Parker was creator as well as finisher.
Cutting in from the right past two tackles, Parker bent a left-foot shot under the bar with goalkeeper Neil Moss well beaten.
Moss made a series of important saves to deny Blackpool further goals, blocking shots from Parker and David Fox with his legs and then in injury time going full-length to push away a shot from Paul Tierney.
Bournemouth were unlucky in the 56th minute when 17-year-old striker Sam Vokes had a shot deflected on to a post before Anderton's dismissal further set them back.
But Simon Gillett, who earlier this season was on loan with Blackpool, scored with a right-foot shot from just outside the area in the 82nd minute.
James Hayter almost equalised three minutes later with a right-foot shot after Steve Fletcher had headed into his path.
But Blackpool's superiority was confirmed in the 88th minute when Fox cut in from the left to beat Moss with a right-foot shot.
Refereee Deadman had a busy afternoon, raising yellow cards in the first half to Bournemouth striker Fletcher and to Parker for questioning a decision.
He was even busier in the second half, cautioning Hayter and team-mate Neil Young for fouls and Blackpool's Forbes as the visitors held on comfortably.
The excitement of the second half was in contrast to the first when both sides struggled to make an impact.
Fletcher went close for Bournemouth in the seventh minute with a volley which was held by Rhys Evans while Hayter also tested the goalkeeper from long distance.