Late goals from Leon Best and Chris Maguire earned the 10-man Owls a 4-2 win at Dean Court after Marc Pugh and Yann Kermorgant had cancelled out earlier efforts from Atdhe Nuhiu and Sam Hutchinson.
The defeat left Cherries three points adrift of the top six with three games remaining – with Brighton moving into the final play-off place following their 1-1 draw at Huddersfield.
Ipswich, who Cherries face at Portman Road on Easter Monday, could open a two-point lead in sixth should they triumph at Watford tomorrow, with Howe set to attend the Vicarage Road clash.
Howe, who watched Cherries play the final 27 minutes against 10 men following Hutchinson’s dismissal for two bookings, said: “We went all out for the win and sometimes there is a danger against 10 men that you commit too many bodies forward. But we had no alternative because we needed to score. I felt we might regret it if we had taken a draw and ultimately proved to be a point short.
“We did everything to try to win the game. We weren’t at our best, there is no denying that, and I didn’t think we managed the last part of the game very well when we had the one-man advantage. The biggest disappointment was that we didn’t take full advantage of the situation.
“I think you have to give credit to Sheffield Wednesday. They played well, made it difficult for us, played a high line and compacted play. They tried to stop us getting our natural rhythm. We were disappointed we didn’t win but, sometimes, you have to give credit to the opposition and I thought they did well.
“For periods of the game, we showed our quality and then, at other times, we were a little too wasteful in possession, which isn’t like us. We couldn’t build any pressure because of that. Although we weren’t at our fluent best, I still felt we did enough in the game to have won but they are the fine margins we have come on the wrong side of in this game.”
Asked whether he felt Cherries could still reach the play-offs, Howe replied: “From my perspective, I felt we had to win all four games so I think it is going to be very difficult but we will try to win the remaining three. With three games left and a goal difference which isn’t too clever, I think it will be diffiuclt.”
Howe added: “When you see where the team was at the start of the season compared with where it is now, I think there has been a lot of progress and a lot of positives. I have been delighted with the players’ attitude and commitment to the games and to their work.
“We desperately want to finish as high as we can. We don’t want the season to fizzle out and there is plenty to play for as far as we are concerned.”
Asked whether he felt Cherries could still make the play-offs, Sheffield Wednesday boss Stuart Gray told the Daily Echo: “I think they probably felt as though they needed to win today and I would imagine Eddie would be very disappointed with losing at home. People may look at this result and think it was a bit of an upset but it wasn’t. I thought we deserved to win.”