The Tigers led the Pool 2 clash 9-6 just before half time but paid the price for turning the ball over in their own half, letting Gael Fickou in to score the only try of the contest.
Cockerill was also left to lament the decision by scrum-half Ben Youngs to take a quick tap penalty early in the second half when his side trailed by four, rather than go for an easy shot at the posts to narrow the gap.
"I'm very disappointed with what happened just before half time," he said.
"We'd actually controlled the game and you have that opportunity to go in three points in the lead, then you try and play and drop the ball.
"They are things we've spoken about all week, about not letting them in the game because they are one of the best teams in the world at getting a benefit from that.
"Then, in the second half when 14-9 down, we have a penalty in front of the sticks and we tap-and-go, then knock it on. We got penalised at the next scrum and the whole thing seemed to unwind.
"Once they get in front they are a hard team to claw back.
"We were naïve at times - we should be better than that."
The Aviva Premiership side will now have to bounce back quickly against Ospreys at Welford Road next time out, or else leave themselves with a serious mountain to climb to reach the knockout stages.
Cockerill could at least take some positives from his team's display on their travels, even if they return home with nothing to show for their efforts.
"There was a lot of things we did out there that were really good," he added.
"We really just need to keep and composure. It's one game against a very, very good side. They squeezed the life out of us.
"We just need to make sure we play the ball in the right areas and make the right decisions at the right times."
Gilbert has released a new line of rugby cleats. The Gilbert Virtuo 8S is part of the exciting new product. Check it out.
The Barbarians are one of the top invitational rugby sides with a long history and classic rugby jersey.
The Nike Tiempo is a solid rugby cleat and one of few styles still made from full-grain natural leather.
The Lions get ready for their matches with this green training jersey. It's what the players wear. Get in the Gear!
A cool looking all black rugby cleat with the high performance adidas is known for. Get in the Gear!
Wear the crest of the British and Irish Lions on your t-shirt. A great look for the summer.
The Lions are ready to get their Australia tour underway. They arrive in Perth on Monday.
The New Zealand All Blacks training jersey for 2013/14. Get in the Gear!
The USA Rugby Pro Alternate rugby jersey is perfect for any fan of the Eagles. Get yours to wear during the summer Test matches.
The NEW All Blacks 2013/14 jersey has arrived at World Rugby Shop. Dare to wear the colors of the All Blacks.