Villa led from a Joe Cole goal in the first half and a frustrated Lambert said: "We should have won it. We should have been out of sight in the first half to be honest. With the chances in the first half, we should have put the game to bed.
"The way we played for long spells of that game was really good. We should have killed the game off. We had chances to do it.
"Burnley came into it to their credit when they got the penalty and it puts you on the back foot. We lost two points. We could have lost three towards the end when Burnley hit the post. Burnley came at us really strongly in that little burst of play.
"When you pick up points away from home, it's good in this league, but this was one we should have won. This is a game that got away from us."
Lambert made one change from the team that had drawn against Southampton on Monday, bringing in Joe Cole. He added: "The big plus was Joe Cole with the way he played. I thought he was outstanding. He was exceptional. I thought he was a massive plus and it was a great goal he scored.
"He's getting fitter and fitter with every passing week. He has great enthusiasm for the game of football for a guy who has won everything in the game. His performance today was outstanding."
Dyche watched his side add this draw to the two wins against Hull and Stoke. "It was a well fought point," he said. "A lot has been made of Aston Villa, but they are a good side and there is certainly no question of them downing tools.
"They are a team full of energy and desire to win a game. I felt they started well to be honest and the feel of the game was going their way, but we calmed it down before half time.
"In the second half they tried to drain the life out of the game but I'm pleased to say we brought it back to life with a big finish.
"In the end it's a strange one, in that we are scratching our heads that we didn't win it. Perhaps though, it's a sign of how far we've come that we are 1-0 down at home to Aston Villa and we're disappointed not to win.
"However, it's a great thing when you are picking up points. We came back from a goal down to secure a point and they are good things for the group. When I asked the lads how they were feeling after, they said disappointed. That's a marker for where they are mentally."
Dyche added: "I take pride that we have so many motivated people and that we can play and affect games in a different way. We are 1-0 down in the Premier League to a good side, so to come out of a lull in the game looking so sharp as the game wore in is excellent.
"I thought we created plenty of chances today and that's another good sign. We definitely had the better hand in that respect and it's good to see the chances we're creating are getting better.
"There is a nice confidence and belief in the team and it was evident. It's also important that the three substitutes have an impact and they definitely affected the game. I speak to them a lot about that and they certainly delivered."