Two second-half tries from centre Wesley Fofana, and a score by Julien Malzieu just before half-time, helped to secure a 30-12 victory for hosts ASM Clermont Auvergne at the Stade Marcel Michelin.
But the teams meet again at Welford Road next Saturday with everything still to play for in Pool 4.
Scrum-half Ben Youngs provided the opening try of the game in Clermont, but Tigers were 25 points in arrears when the second score came courtesy of a penalty try with 10 minutes remaining.
“We did some good things, especially in the first half, but we got punished when they created openings,” said director of rugby Cockerill.
“It is disappointing because we obviously wanted to take something out of the game here.
“Parts of the performance were very good, the set-piece went well, and I thought Marcos Ayerza and Martin Castrogiovanni were outstanding in the scrum for us.
“We had two or three opportunities, especially in the opening 20 or 25 minutes, but did not take them. They defended well and there was not a lot of quick ball in the game.
“Clermont played well. They kicked a high one we didn’t deal with and they got the score before half-time which hurts us. Then we had two guys in the bin in the second half and the momentum was against us.
“They have quality in the backline and if you give them some ball and some space in broken field then they are going to punish you, and they did that.”
Manu Tuilagi made his return from injury, making only his second appearance of the season for the club following his World Cup campaign.
“He’ll be better for that game,” said Cockerill. “He’s had a full game, and Ben Youngs too has had another game under his belt.”
A capacity crowd of 18,000 saw a full-blooded encounter at the Stade Marcel Michelin, and an even bigger crowd is in prospect for the return fixture on Saturday.
“To come here, in a hostile environment, a sold-out game, it was a great place to play,” said Cockerill who spent two years in Montferrand as a player.
“But we’re disappointed we weren’t good enough to win on the day.
“Now we’ve got a short turnaround and I’m sure there’ll be a great atmosphere back at Welford Road for another massive game on Saturday.”
