The Bees went ahead after an uncharacteristic mistake by Wednesday keeper Keiren Westwood who let Nico Yennaris' shot slip through his hands in the ninth minute.
However, after dominating the majority of the first 45 minutes, Brentford would be going in at the break on level terms after Gary Hooper's equaliser on the stroke of half time.
Wednesday rallied and were much-improved in the second half and they found a deserved breakthrough, on the balance of play at that stage, on 70 minutes through Ross Wallace.
A frustrated Smith admitted that missed opportunities were his side's downfall.
"After the first half, we should have been walking off the pitch three goals up," he said. "We didn't take our chances, though, and we then had that sucker punch at the end of the first half.
"We have got to learn as a team, and a lack of nous has cost us the equaliser. We should have had the game sewn up.
"They started the second half better than we did and until they scored we huffed and puffed a little bit. We were then the better team again after they went in front and deserved to score another goal, but it wasn't to be. We have to keep believing in what we are doing."
Smith said that Wednesday's recovery after the break was a combination of the Owls improving and Brentford suffering the psychological blow of an equaliser just before half time.
"It was a bit of both," he added. "They were buoyed by the goal, they probably feel themselves they probably should have been out of the game with the chances we created. It was probably a sense from our lads as well of where we should be and were we actually are in the game.
"When you haven;t won for the first six games a little nervousness creeps in. Performance-wise I can;t ask for any more but unfortunately we switched off from a corner lost us the game but not taking chances cost us the game."