Villa moved ahead towards the end of the first half at Turf Moor, Joe Cole firing the visitors into a deserved lead.
Paul Lambert's men cut through the Burnley defence on numerous occasions, but were often let down by wasteful finishing.
And Villa's winless streak in the Premier League was extended to nine games courtesy of Danny Ings' 87th-minute penalty, awarded after Jores Okore hauled down Lukas Jutkiewicz.
Burnley could have claimed all three points in the dying stages as Jutkiewicz and Ings both went close, but Dyche was proud nonetheless.
He told reporters: "Ian Holloway said last year that we have a strong chin and it's harder to do that [come from behind] in this division of course, because you can get punished.
"But it's fair to say that was on show again. The energy, the quality - the mix today was right, particularly in the last 10-15 minutes when that's when you think we should have won it.
"I've been pleased with all of the group. It's a combined effort and I think that was on show.
"I'm enjoying what the players are doing anyway, but certainly the last two games [which have brought wins over Hull City and Stoke City] and today because the mentality is quite clear, but also the quality.
"I thought some of the play [against Villa] was excellent and some of the chances were real golden chances, and that's an important factor. We just need to take more of them."
