Kevin-Prince Boateng was outstanding in the 3-0 win.
"We went through the qualifying round and will play the Champions League, so tonight we concluded the comeback that we started in Serie A in January," a visibly annoyed Allegri told Sport Mediaset.
"The lads fought hard, though PSV are a good team with great technique and we struggled at the start of the second half.
"There was some tension and anxiety, but the lads defended well and we only ran a few risks after the break."
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