Second-half strikes from Ross Barkley and Romelu Lukaku were enough for the Toffees to progress, but Howe felt that his side still had enough chances to recover.
"We were well in the game," he said.
"We were very dogged and the lads gave everything again - I have no criticism in terms of commitment or lack of effort.
"I think Everton just had the extra quality and, of course, at key moments in the game we didn't take our chances again."
The key chance for Howe's side came after 37 minutes, as Charlie Daniels stepped up to take a penalty after Jake McCarthy handled a cross in the area.
But the left back was denied by Joel Robles before Simon Francis then emerged from a sea of Cherries shirts to strike the rebound into the side netting.
"There is no blame attached to Charlie," Howe continued.
"He has been really good from the spot for us - that's the first he has missed.
"It looked like if we weren't going to score the original kick then we would from the rebound, but it's just one of those things. All of the players were eager to get on the end of it."
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