Tigers finished runners-up in Pool 5 and now face a last-eight trip to the 18,000-capacity Stade Marcel Michelin where ASM Clermont Auvergne have an unbeaten home run stretching more than 70 games.
Director of rugby Cockerill, who spent two years at Clermont as a player, said: “We’re still in the Heineken Cup and we’re still in there fighting.”
Ulster won 22-19 in a colossal encounter at Welford Road on Saturday to top Pool 5, with scrum-half Ruan Pienaar scoring all of their points, including a chargedown try.
“We’re obviously bitterly disappointed,” said Cockerill. “It all comes down to tiny margins in the end and unfortunately we’re on the wrong side of them this time.
“Both sides gave a massive effort, everybody worked very hard, and it was always going to come down to small margins. I’m not going to criticise the lads, Ulster played very well, we played very well, it was a very tight game.
“We got into a good position at 10 points up, but Ulster are a very good side, it could have gone either way and in the end we got beaten.
“We scored a nice try and conceded a poor one. It’s a great nudge from Floody for the Niall Morris score and a great kick to convert it. We got 10 points ahead but then gave those points back too easily.
“I thought both sides defended very well and Ulster didn’t really create an opportunity to score but they kick well and put you under pressure. They are a very tough side.
“We were good enough to win but I’ve got no complaints, fair play to Ulster, they are a very good side, both teams played well. You give congratulations to Ulster and commiserations to the Leicester boys.”
Clermont clinched a home quarter-final with a bonus-point win over French rivals Racing Metro on Sunday and will face Tigers in the tournament for a third time in five seasons.
“We let an opportunity slip to have a home quarter-final but these are one-off games and we are certainly not defeatist,” said Cockerill.
“We know the challenge but we will go anywhere and give it our best shot. Just because we’re in an away quarter-final doesn’t mean we think we cannot win.
“Last year we went to Toulon in a massive battle and just lost by six points. We will have a few bodies back by then, we go wherever we have to go and try to win.
“We’d have preferred a shorter trip – to Welford Road – but credit to Ulster for what they have done and we live to fight another day.”
Injuries to Mathew Tait (back), Graham Kitchener (ankle) and Jordan crane (shoulder) forced line-up changes during Saturday’s match and all three will be assessed this week.