Lee had the first attempt in the early minutes of the game, but Brentford keeper David Button was well placed to deny the midfielder.
Andre Gray found himself through on goal with just over 10 minutes of the first half to play, but struck his shot wide.
After the break, Gray went close once more, cutting inside before narrowly striking wide with Wednesday keeper Kieran Westwood at full stretch.
Both teams struggled to take a hold on the match, and Caolan Lavery saw his attempt blocked as Wednesday looked to get the opener.
With 15 minutes to play, Wednesday broke the deadlock after substitute Sergiu Bus found Lee, who tapped the ball into the net after good initial play from Lavery on the wing.
Alan Judge almost equalised for Brentford, but shot over the bar as the visitors failed to break down the Wednesday defence.
The loss leaves Brentford seventh in the table, with Ipswich three points ahead in sixth.