Cockerill's team were smashed 41-7, surpassing their previous record European defeat - 33-0 against Ulster in Belfast eight years ago.
The twice Heineken Cup winners are now facing only a fifth group-stage exit in 15 European seasons, with Cockerill believing they will finish third behind pace-setters Ulster and Clermont Auvergne.
Leicester cannot win Pool Four - which offers an automatic quarter-final place - after Ulster moved seven points above them ahead of next weekend's probable group decider against French heavyweights Clermont.
Tigers will need a bonus point victory over Italian strugglers Aironi at Welford Road to give themselves any chance in the race for two best runners-up spots, yet teams in their group and other pools appear better placed.
But Cockerill said: "We are out. We will be third in our group. We are done."
Leicester were without injured England trio Toby Flood, Manu Tuilagi and Louis Deacon, a fact not lost on Cockerill.
"It shows in European competitions with the English sides having a limit on what they can spend that if you have key guys missing, then it is going to very difficult for you," he added.
"We've got some key guys missing, but that is not an excuse and I don't want to use it as an excuse. We should have played better tonight.
"They (Ulster) played very well, and you have to take your hat off to them, shake their hands and say 'well done'.
"We struggled to get a foothold in the game. They controlled the field position, took their chances and built the score. There wasn't a part of the game where we had dominance.
"You don't want to be conceding 40 points. They are a good side, and we didn't cope with it. We never got on the front foot enough, and when we did, they worked very hard on the ball and we couldn't cope with their intensity.
"We needed to come here and get a result, and we thought we could do that. The side we had out tonight was good enough to put up a better show than that."
Leicester were undone by two first-half tries from Ireland international wing Andrew Trimble, while Ulster's South African scrum-half Ruan Pienaar put the boot in with five penalties and three conversions for a 21-point haul before wing Craig Gilroy and replacement Paul Marshall added late tries.
Skipper Geordan Murphy claimed an early touchdown for the Tigers, converted by fly-half Billy Twelvetrees, but Leicester did not score after the 16th minute as Ulster dominated.
Ulster's South African skipper Johann Muller said: "That was definitely one of our best performances in terms of taking our chances and the passion we showed.
"Ravenhill is always a good place for us to play, and the atmosphere tonight was outstanding. The boys delivered a great performance.
"We said at half-time that the bonus point was on, and it was about being patient. We knew it might take 81 or 82 minutes, and the boys never stopped.
"But it doesn't mean anything unless we get some points in Clermont. We have to make sure we do what we have to do."
And Ulster coach Brian McLaughlin added: "We have a pack of forwards that is starting to be the envy of any team in Europe. The scrummaging and work around the pitch tonight was superb."
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