The hosts took the lead after just 13 minutes courtesy of a fine Justin Hoyte strike. Emanuel Ledesma fed a clever ball through to the right-back, who smashed his shot first time with the outside of his boot past Chris Kirkland.
Josh McEachran almost made it 2-0 just after the half-hour mark when he danced through the Wednesday defence, only to see his effort saved by the feet of Kirkland.
The visitors tested Jason Steele at the end of the first half, and started the second half in similar fashion, equalising just four minutes after the restart when Jay Bothroyd and Michail Antonio linked up to allow Gary Madine to finish, although replays suggested that the goalscorer was offside.
The hosts all but wrapped up the three points when they went 3-1 up with a little under 15 minutes remaining, Lukas Jutkiewicz finding himself unmarked at the near post to nod in Grant Leadbitter's corner.
Wednesday had a chance to pull themselves back into the match in the 87th minute when Madine found himself through on goal, but Steele was equal to his effort, allowing Boro to see out the closing stages in comfort.