Mauro Boselli's 18th minute goal was enough to settle the contest at Dean Court and book the Premier League club a fourth round date with non-league Macclesfield.
But Martinez, who only named Emmerson Boyce and Maynor Figueroa from the team who drew with Fulham at the weekend, knew his players had to work hard to end Bournemouth's 18-game unbeaten run under Eddie Howe.
"It was a very difficult game as we expected," he said.
"We know that Bournemouth have been unbeaten and it doesn't matter at what level you do that at because you have got a very strong unit.
"I must admit Bournemouth deserve huge, huge credit over the two games in the manner they play football and I think it is going to bring real success to this football club.
"We knew that we had nothing to win today and had to be very professional; really, really strong and I couldn't be more proud of my players. When you make nine changes, it is not easy."
When asked if he was taking a chance by naming an inexperienced side, Martinez said: "No, it is not a risk. A risk would be not to give the opportunity to youngsters."