Benfica won Thursday's quarter-final second leg in Lisbon 2-0 to triumph 3-0 on aggregate with Rodrigo scoring a brace for the home side.
Advocaat paid tribute to Alkmaar's work ethic in Portugal but claimed his team were simply inferior compared to Benfica.
"We have reached the last eight, I have given the boys a compliment," Advocaat said.
"We have come far and have tried also against Benfica to make a contest.
"Qualitatively, however, we are of course a lot less."
The 66-year-old coach added his team 'tried everything', while Alkmaar midfielder Steven Berghuis was left frustrated that the Dutch club had been unable to make a mark in Lisbon.
"I stand here with mixed feelings," the 22-year-old told AZ's website.
"Of course we can be proud that we have come so far in the Europa League but I hate that we did not bite here."
Berghuis was fairly happy with how Alkmaar kept their shape in defence but bemoaned the fact the visitors had conceded two goals on the counter-attack.
"The two goals we conceded were almost identical," Berghuis said.
"Both times they came out quickly after we lost possession. Twice there was a deep cross from the right and both times Rodrigo tapped the ball in at the far post.
"Handsome attacking from Benfica but we should have anticipated better as a team."
The loss in Lisbon stretched Alkmaar's winless streak in all competitions to five games and, having been eliminated from the KNVB Beker late last month, Advocaat's side now only have the Eredivisie to focus on.
Alkmaar sit seventh in the Eredivisie with three games left before the Dutch play-offs start and their boss quickly turned his attention to their league clash with NEC Nijmegen on Sunday.
"We must ensure that we make the play-offs. Sunday against NEC is again an important game," Advocaat said.
"We need to be fit and sharp."
The top eight clubs will reach the play-offs and Alkmaar have a seven-point gap on ninth-placed PEC Zwolle ahead of the weekend's games.
