Dean Smith's Villa side created enough opportunities to win the match and had 27 shots but only got seven of them on target and Burnley keeper Nick Pope dealt comfortably with most.
The game was surprisingly open with Villa's Jack Grealish and Burnley's Dwight McNeil both able to show off their dribbling skills, but the Clarets will feel fortunate to have headed North with a point.
"It was a performance worthy of a win, we had enough chances to win three games. We were so good, 27 attempts to six shows that," Smith said.
Villa hit the woodwork twice in the first half with Anwar El Ghazi's free kick superbly tipped on to the bar by Burnley keeper Nick Pope before Ahmed Elmohamady headed against the post from a corner.
After the break, the influential Grealish created plenty of openings for Villa but El Ghazi wasted two good chances, weakly shooting straight at Pope on both occasions.
Villa defender Tyrone Mings had a good chance late in the game but headed wide after he was left unmarked in the box at a free kick.
The result left Villa in 11th place on 19 points while Burnley move out of the bottom three into 17th spot on 10 points.
Burnley are unbeaten in their last three games including Sunday's win at Arsenal and manager Sean Dyche was relieved to keep that run going.
"I am pleased with the point overall. They edged it with the chances and the quality of chances," he said.
"When your two away games are Arsenal and Villa you will take four points and two clean sheets".