After topping a pool featuring Toulouse and the Ospreys, Tigers had to travel to the Stade Mayol to face French league leaders Toulon in the quarter-final but headed home after a 21-15 defeat.
Jonny Wilkinson kicked all of the Toulon points, with six penalties and a drop goal, with Toby Flood replying for Tigers.
A desperately disappointed Cockerill said: “It was one big, hard, physical game. Either side could have won. It was a game of tiny margins and they got those margins in the end.
“It was a great contest, a good game from both sides. We pushed them all the way to the wire and they are a good team. They won today, we accept that, but we could have easily won.
“We were under pressure for parts in the second half but the effort from Leicester players was phenomenal.
“It is hard place to come and we knew that. Toulon are top of the Top 14, one of the hardest, most physical leagues in the world. They are a big, physical side and when they came at us in the second half, the effort and the ‘bodies on the line’ from our guys were second to none.
“Sometimes you just see the effort, the attitude and the way we played – the mindset was fantastic – and you have to applaud that in the players. Credit to them.
"The support was fantastic too.
“You get that bounce of the ball at the end and we could have won the game.
“It’s hard and raw at the moment but sometimes you lose and we have to deal with it.”
Tigers led 9-0 inside the opening 20 minutes but lost fly-half Flood to a yellow card shortly before half-time and then prop Dan Cole to the sin bin in the second half, both for deliberate knock-ons.
Toulon took the lead for the first time in the 47th minute and closed out the game with a Wilkinson drop goal in the last minute.
Tigers lost Tom Croft to a back problem just before half-time and prop Marcos Ayerza to a collarbone injury in the second half.