The two-time European champions progressed to the semi-finals of the continent’s elite club competition once again but were made to fight hard for it.
“They’re a good team, we’re a good team, someone’s got to lose,” said Cockerill after the game.
“There are tiny, tiny margins, even for their first try I thought their player knocked it on. It’s tiny margins from officiating. There’s all those things – there are a million things that can happen in a game.
“Munster are a good side but so are we – two very good teams going very hard at the game and it was a great contest. So I’m very proud of how we performed.”
Edinburgh will now focus on finishing strongly in the Guinness Pro14 to reach the play-offs and ensure the return to the top-tier Heineken Champions Cup again next season.
Keith Earls scored two tries, while Edinburgh centre Chris Dean crossed in the hard-fought battle.
“Did they deserve to win? Well, they won, so well done to them,” added the home coach. “We’ve got to learn how to win these tight battles, and we didn’t do it today. You watched the game – there was nothing in it, was there?
“We could have easily won that game, and we’ve got to find that belief. I think we have to a point, but in the tightest of moments we’ve got to be better. We’ve just got to keep working at our game. We’re two years in to being well coached and having a good process and a good environment. Munster are probably four or five decades into that.”