Taylor almost doubled Millwall's advantage after 30 minutes, but the Wednesday keeper was able to tip his free kick over the crossbar.
Kirkland had to prevent the Owls from slipping behind again shortly after the re-start, as Taylor drifted in a dangerous-looking corner, which the keeper claimed before Millwall could took a shot.
The Wednesday goalkeeper was kept busy throughout the second half, as the home side created the better of the chances, but Kirkland was able to keep the scores level.
Llera was on hand to make a late clearance off the line, before the visitors took the ball down the other end of the pitch, Chris Maguire volleying in deep into stoppage time to seal the points for Sheffield Wednesday.
Wednesday jump up to 14th in the Championship table as a result of the victory, while Millwall slip to 16th.