Millwall had a chance to get themselves back into the game on 24 minutes when Karleigh Osborne found space deep inside the area, but he couldn't keep his shot down.
The visitors drew level on 35 minutes, when James Henry latched onto a pass from Andy Keogh and found the top-left corner.
Less than 10 minutes later, Wednesday found themselves 2-1 down when Liam Trotter arose in the box unmarked and fired home.
Wednesday came out all guns blazing in the second half, with Johnson almost doubling his tally for the afternoon on 50 minutes and Rodri shooting narrowly wide shortly after.
Their efforts paid off on 68 minutes when Llera slid the ball past Taylor from close range to level the score at 2-2.
Wednesday looked odds-on to clinch a winner. Rhys McCabe was a live wire in the closing stages, forcing a reaction save from Taylor, while Johnson came close again on 80 minutes.
Jones's men left it late, but they put the game to bed in the end. Llera was the hero of the afternoon, firing the ball into the top corner from inside the six-yard box to give Wednesday a 90th-minute winner.