Steve Cook’s injury-time winner earned Cherries a 1-0 victory at Barnsley and saw them climb a place to 11th in the Championship – still 10 points adrift of sixth.
Howe said: “We would practically have to win every game which, at this level, is incredibly difficult. But while there is a chance, we will give it everything we can to see if we can get there. And if it isn’t this season, we will try to build and be better prepared for next season.”
Asked whether he felt Cherries would have exceeded expectations had the season already finished, Howe replied: “I would say we would have achieved our objective and that is how I would describe the season if it were to finish now.
"We would have achieved what we had wanted to do so you would have to say it had been a success. There is no denying it would have been a success if it had finished now and we are in a good position to move forward with a young team to try to get better next season.”
Discussing the win at Oakwell, Howe said: “Barnsley started well and had a couple of chances. But, overall, if you look at the game as a whole, I felt we were the dominant team and created more chances.
“But Barnsley are fighting for their lives and gave everything. It was hard on them and we know how they felt because we had the same on Tuesday. We lost to the last kick of the game at Charlton and it is a cruel one. I feel for them but I was really pleased for my team and our supporters.
“Football is a strange game. We were so low after Tuesday when we felt we had done enough to get another important point. To concede with practically the last kick of the game was a real sickener and then we did the same to Barnsley.
“I felt we deserved it. I don’t think there was any doubt about who had been the better team but we didn’t score when we should have and it took a set-piece at the end.
“It pleased me no end to get another clean sheet because that isn’t something we have been renowned for this season. But I think that has been the real improvement in the team in recent weeks.”