Dan Gosling opened the scoring for the Cherries in the first-half before second-half strikes from Charlie Daniels and Benik Afobe sealed a resounding win at Vitality Stadium.
Howe was delighted with the result, not because his men beat a side around them in the table, but because of the players' character to bounce back from Tuesday's 3-1 defeat to West Ham United.
"It's a big step forward on the back of Tuesday night's game," Howe said.
"We didn't play well against West Ham and didn't get the result, but the players responded today and their true character was there for all to see. Today was the opposite of Tuesday, we were solid defensively and got goals at good times.
"We didn't get the second goal we needed against West Ham but we got it today.
"It wasn't the complete performance but I'm very pleased with the players today, we started positively and it continued right throughout the game."
Howe understandably praised Benik Afobe after the striker netted his first Premier League goal, but was also quick to praise the contribution of Marc Pugh who was instrumental in the opening two goals for the Cherries.
"It's really important new players come in and do well," Howe added.
"Benik is a striker and it's important for him to score goals. The finish was a high quality from a great ball from Charlie, he did very well today and made some good runs so there's good signs for him.
"I don't have to credit Marc Pugh for his attitude and application, he's the model professional. He's ready when needed, did well in the FA Cup last week and I'm so pleased for him for his tireless work."
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