Stankovich, 32, saw red with 16 minutes left for a high tackle on scrum-half Darren Allinson in a match Leicester won 22-15 to keep alive their hopes of retaining the cup, having lost their opening Pool 2 game at Saracens.
Tigers director of rugby Cockerill said: "It was a bit harsh. I really feel for Boris because he is not a dirty player. It was a yellow at most.
"Boris has gone with his arms against a smaller player and the guy slipped into him. I think the referee was the only person in the place who thought it was a red.
"If it goes to disciplinary I hope the sending off is sufficient because Boris will be important to us in the next couple of weeks.
"It was a disappointing decision but the lads got on with it. We conceded a try and the momentum went against us but our defence was outstanding."
After Stankovich's dismissal Irish, trailing 19-8, staged a ferocious fightback and a try by Tom Homer, converted by Ian Humphreys, made it 19-15 with just 11 minutes left.
But Leicester held out and fly-half George Ford's fifth penalty sealed victory for Leicester.
Reports suggest Ford, 19, is considering his future due to a lack of first-team rugby but Cockerill said: "George is considering his future because he is out of contract, nothing new there.
"I don't think it's anything to do with a lack of action. He started 11 games last year, some of them massive. He has started three so far this season. He is still learning. We want him to stay and I have every confidence he will stay."
Cockerill was delighted with a win earned by what was effectively his second string side against a strong Irish outfit.
"We said before the game that they will look at our team sheet and expect to win, which is why I'm surprised they picked the side they did," he said.
"We had 22 guys unavailable and we said we could win but we had to be really good. The lads should be proud of that result, and we are back in the competition."
Irish scored two tries through Anthony Watson and Homer, to Leicester's solitary effort by lock Rob Andrew.
Irish boss Brian Smith said: "We are disappointed. We got a decent start and if you look at the teams you would say London Irish had more talent on paper.
"But you don't win games on paper, you win them on the park and Leicester were very enthusiastic. They played up to their potential.
"We outscored them two tries to one, which is an achievement at Welford Road but we got absolutely hammered with penalties, 14-7. That killed us."
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