Irish looked on target to record a famous victory when they led 32-31 with two minutes to go, but Toby Flood's penalty and a Tom Croft try from the last movement of the game proved heartbreaking for the hosts.
"We played some good stuff but some horrendous stuff," Cockerill said.
"We soaked up early pressure and then I thought we were going to take control, but we were too inaccurate against a side who have the pace and power to punish you."
Leicester certainly had the dominant scrum and Flood's last-ditch penalty, which give him an impressive match tally of 26 points, came from Irish conceding a penalty in that area.
Cockerill said: "We were aware that their scrum had been weak all season.
"We came with a plan to dominate the set piece which we did, but apart from that we didn't play particularly well."
Leicester did lose Ben Youngs with a minor shin injury midway through the second half, but the England scrum-half may have further problems as he could be cited for what appeared to be a deliberate kneeing of an opponent while he was on the floor.
Irish coach Toby Booth described Youngs' actions as "unnecessary" and Cockerill added: "It was in the heat of the battle.
"He was being held but it is a matter for the citing officer."
Irish were desperately unlucky not to have secured at least a share of the spoils after being the more creative side, with Tom Homer especially unfortunate to be on the losing side.
The full-back scored a splendid try and kicked three late pressure penalties to put his side in sight of victory.
Irish director of rugby Brian Smith said: "Homer is a special kid and it was a special performance.
"As a neutral the game was an outstanding spectacle. After 78 minutes we had our noses in front and I thought it was going to be a Cinderella moment for us."
Booth added: "We should celebrate the effort, there were a lot of positives, but ultimately it's all about the four points."
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