Brett Pitman's fifth goal in as many matches took his tally to 15 since returning to the club from Bristol City.
Pitman's goal came in the 11th minute with a volley after Marc Pugh and Matt Tubbs had opened the Scunthorpe defence.
After watching the Cherries claim their fifth successive win, four of them without conceding a goal, said: "This was not our best performance. We were well in control but we could not turn our possession into chances and goals.
"We created enough to have by a bigger margin but it was always going to be hard against a team fighting for their lives.
"Our problem is that we have played more matches than most of our rivals so I think we must win all four of our last matches.
"I have every faith in the players and I would not ask it of them if I did not think it was possible.
"Scunthorpe outnumbered us in midfield but we still had far more of the ball and I do not think there is any doubt that we deserved to win."