The Dutch side stunned the Santiago Bernabeu by winning 4-1 in midweek to reach the quarter-finals.
Ajax ran riot despite trailing 2-1 from the first leg in Amsterdam as the crisis at Real Madrid deepened.
Hakim Ziyech gave the visitors the lead before David Neres doubled their advantage inside 20 minutes.
Things got worse after half-time when former Southampton man Dusan Tadic made it 3-0.
Marco Asensio pulled one back for Real Madrid to spark a late comeback attempt which was crushed shortly after by Lasse Schone’s sumptuous free-kick.
Brendan Rodgers’ men crashed out in the third qualifying round to AEK Athens back in August.
Before his controversial departure to Leicester, they were beaten 3-0 by Valencia on aggregate in the Europa League Round of 32.
“Brendan Rodgers talked about the glass ceiling for Celtic in Europe but Ajax have just smashed that theory to pieces,” Sutton wrote in his column with the Daily Record.
“The Dutch side’s incredible victory against Real Madrid in the Bernabeu this week wasn’t just one in the eye to the UEFA bigwigs who want to make the Champions League an elite team closed shop.
“It was also a message of hope to all sides that money isn’t everything, even when it comes to the millionaire playground. Ajax are not even top of the table in Holland.
“Celtic fans have been constantly told the club haven’t got the resources to cut it at the very top level any more but interestingly, Ajax have a wage bill that is more than £10m less each year.
“Their turnover was £20m less the Celtic’s last season. Talk about getting it right.”
Rodgers faced strong criticism when leaving for the Foxes including from Sutton, under the narrative that the Bhoys are a bigger club than the Premier League side.
One of the statements from that argument was that Celtic play in Europe, but given their lack of success in the competitions, it perhaps doesn’t hold as much pulling power as some suggest.
“This is the model Celtic should be aspiring to replicate in Scotland,” Sutton added.
“There is no excuses for spending £59m on wages and not getting value for money and it goes back to the structure throughout the club.”