Midfielder Dan Gosling gave the hosts the lead after 10 minutes with a close-range header from Marc Pugh's cross and Charlie Daniels doubled the advantage with a penalty nine minutes into the second half.
Afobe, signed from Wolverhampton Wanderers last week, wrapped up the points 15 minutes from time and Bournemouth moved above Norwich in the table.
"It was the perfect afternoon in many ways. Any win in this league is so valuable and the players produced an excellent, assured performance," Bournemouth manager Eddie Howe told the BBC.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
"We knew the importance of this month going into it and today is a big step. We've got Sunderland next week and the games just keep getting bigger."
(Reporting by Ed Osmond; editing by Clare Fallon)
