The Owls were hunting all three points to boost their chances of avoiding the drop to League One and took the lead after just four minutes thanks to a Julian Borner header.
Things got worse for the visitors when Henri Lansbury was given his marching orders in the 21st minute and the Robins were up against it.
Sheffield Wednesday looked on course to clinch victory, but just three minutes from time Tyreeq Bakinson levelled with a right-footed effort from outside the box, which went into the bottom left corner of the net.
It ended 1-1, meaning the spoils are shared, and Owls assistant Smith admits that the mood is one of disappointment.
“It’s a tough one to take, feels like a defeat, it’s a point but not enough”, Smith said post match.
“We should have won the game. We had enough chances to win the game.”
Sheffield Wednesday sit in 23rd in the Championship table and seven points off safety, with just four games left to play, presenting a mountain to climb.
Smith though has vowed the Owls will keep going and focus on their next game.
“We can’t just give up, we have got to work. But we have to take chances when they are there.
“It was a big game today that we had to win, we didn’t and we have to move on to Tuesday.”
Next up for Sheffield Wednesday is a home meeting with Blackburn Rovers.