Bale's wicked cross was aiming for Hal Robson-Kanu, but the Wales attacker was beaten to it by Northern Ireland's Gareth McAuley, who turned it into his own net in the 75th minute as Michael O'Neill's men went crashing out, meaning Wales will face either Hungary or Belgium in the last eight.
Coleman accepted that his team did not perform as well as in their 3-0 win over Russia in the group stage, crediting Northern Ireland for making it hard for Wales and praising Bale for making the difference.
"There was so much riding on this game," said Coleman. "We got a bit of luck with the goal we scored, it was unfortunate for big Gareth McAuley, but the luck came our way.
"We could have lost our way because the performance was not the same as the Russia performance. We could have become ill-disciplined and lost our shape but we hung in when we needed to.
"That gave us a little bit of time. With Gareth Bale or Aaron Ramsey on the pitch, a little bit of brilliance can change your life and we were depending on that today.
"I thought we didn't really play well at all today. That wasn't our fault, it was credit to Northern Ireland who did what they do very well - make it very difficult for the opponents."
Coleman added that while it was an ugly victory for his side, he is pleased they have found different ways to win matches throughout the tournament.
"I'm not going to take any credit away from our boys," Coleman said. "It was an ugly win for us but who cares? We've played very well and not got results.
"The three wins out of four we've had we've had to play differently. It's a good return for any team."
Northern Ireland coach O'Neill also won praise from Coleman.
"Michael O'Neill is doing a fantastic job," Coleman said. "They'd lost two out of 15. It's a fantastic record for a country like Northern Ireland.
"They'll be disappointed but they've done brilliantly in the tournament. I wasn't looking forward to this game, I knew it was going to be a tough one. They played well today. We never played as well as we could but that's credit to Northern Ireland.
"We will never look past the next challenge. We're sitting here today. Lady luck shone on us but I think we earned it.
"Whoever comes in front of us - Belgium and Hungary can cause us different problems. But we'll worry about that tomorrow and enjoy>