The Cherries were beaten 2-0 at Dean Court thanks to goals from Victor Moses and Daniel Sturridge, but Howe believes that his team gave a good account of themselves against their Premier League opponents.
"We kept going, we never gave up and our heads never dropped – and that really pleases me," the FA's official website quotes him as saying.
"I've just said to the players that if we replicate that level of performance in our own league we'll have no worries about what we're going to do this season.
"I'm very proud because it takes a lot of hard work to get to this stage. It does show that if you work hard, live your life well, then the reward is there. The challenge now is to produce that level of performance in their own league week in week out."
Bournemouth are currently 16th in the Championship table.