The Tigers took a considerable amount of time to subdue the battling Italians at Welford Road on Saturday and led only 9-6 at the interval before four second half tries gave them a comfortable victory.
Clermont's narrow home win over Ulster ensured the Tigers finished in 3rd place in Pool 4 and Cockerill, the Leicester coach said afterwards "The game reflected the importance of it.
"We couldn't see Clermont losing without a bonus point so we knew there was very little hope of us progressing. We didn't get going and it was a pretty average performance."
The Leicester forwards - with excellent performances from Tom Croft, Thomas Waldrom and Ed Slater - enjoyed ample possession but behind the scrum Leicester badly missed the defence-breaking skills of Toby Flood and Manu Tuilagi.
Cockerill admitted: "A number of our boys had a good run out and nobody is injured but we tried to force the game too early and made too many errors in our passing."
Cockerill reported that Billy Twelvetrees will be leaving the club for Gloucester at the end of the season with the Leicester coach revealing: "On Monday he told me he was staying, on Wednesday I was told by his agent that he was signing for Gloucester, who have assured him of being first choice number 12 for them and his salary has been much improved.
"I did not want to lose him and he will be fully involved until the end of the season."
Aironi coach Rowland Phillips was pleased with his side's performance.
"After last week's embarrassing defeat (82-0 to Clermont), it was important that we got our season back on track as last week ripped the heart out of our Heineken campaign," he admitted.
"We never discussed winning but we should have been level at 6-6 at the interval and after half-time we switched off to give them two tries in the first eight minutes."
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