Forest's equaliser, from the penalty spot, came in the 97th minute which left Davies relieved but after the game he said: "I have to apologise to our fans for having to watch that type of football.
"It wasn't a good game, there's no doubt that we didn't play at the tempo or quality we expect of our players.
"Having said that, Burnley had an extra 24 hours and we were forced into making changes because of that. Sam Hutchinson wasn't in good condition after playing his first game in six months against Brighton and Adlene Guerdioura had concussion.
"We also tried to freshen up the two strikers because two games in three days is difficult. All six were forced changes because of the small gap between the two matches.
"I felt quite confident we would get something, even at the end. This team never gives in, as was proved again today. Turf Moor is a very difficult place to come and it's pleasing to go away with a point."
Dyche, yet again was unhappy at a decision and this time it was the time keeping of referee Nigel Miller who had another poor game.
"It was an incredible turn of events," Dyche said. "The goal with the last kick of the game was well over the allotted time, but what can we do? I was amazed by some of the decisions out there, but we don't just have to beat the eleven out there at the moment and it's happening too often.
"It was a penalty, and I've no issue with that, but the fact that it was at 5 minutes and 45 seconds, when only five went on the board, is the incredible thing.
"It's important the fans hear my views about everything, about what we do and I really don't want to be stood talking about officials. I want to be talking about the team and the events of the game, but we are on an impossible run of decisions from officials at the moment.
"We have only had one win in six and in there are four draws. We have all seen that three of those other games should have been wins, but they have been taken away due to decisions we cannot do anything about and that is hard to take."
Having spoken to the referee, Dyche added: "He said he felt the allotted time was correct and that he was going to put it in his report that we were late for kick off, which is interesting.
"I love this club and I am just so frustrated for our fans. We were more than good value for all three points against an in form Forest side and they have seen their team give everything. But we can't get over the line because of things that are taken out of our hands and that is really difficult to take for all at Burnley Football Club."
He concluded: "The crowd were great today. They stayed with the team, which is important. We wanted to go out and be on the front foot, show plenty of endeavour and moments of quality. We just about did that and we can play better still, but as I suggest, to be taken out of your hands is a really tough one.
"We dare to dream about what we can do, but if you look at the points we've had taken away we would be glimpsing up and saying 'well, let's have a go.' Instead we are having to work hard for different reasons, but work hard we will and we will get the points on board."