It looked like a festive season to forget for Villa was set to continue after dropping points at home for the first time this season to Sheffield United and blowing a 2-0 lead to lose 3-2 at Manchester United on Boxing Day.
But Luiz's spot-kick that crept in off the underside of the bar sent Unai Emery's men level on points with Liverpool at the top of the table and break Burnley hearts.
Leon Bailey's opener for the home side was quickly cancelled out by Zeki Amdouni.
Moussa Diaby restored Villa's lead before half-time and they had a man advantage for the final 35 minutes after Sander Berge saw a second yellow card.
But Villa's high line is begging to be picked off by opponents and Lyle Foster timed his run to blast past Emi Martinez 19 minutes from time.
There was a final twist though as Jhon Duran was upended inside the box and Luiz fired home.
- Man City cruise -
Despite missing Erling Haaland for a seventh consecutive game due to a foot injury, City were never troubled by the bottom-of-the-table Blades.
Rodri ended Sheffield United's resistence after just 14 minutes when the Spanish midfielder surged forward and fired low into the far corner.
Julian Alvarez again ably filled Haaland's shoes as the Argentine tapped in his fourth goal in three games to make the points safe from Phil Foden's cross just after the hour mark.
City move alongside Arsenal on 40 points, just two points behind Villa and Liverpool, who are in action at home to Newcastle on Monday.
- Palmer inspires Chelsea -
Chelsea snapped a run of four consecutive away defeats thanks to Cole Palmer's double but had to withstand a nervy finale to beat Luton 3-2.
Powerful strikes from Palmer and Noni Madueke gave the visitors a 2-0 half-time lead.
Mauricio Pochettino's
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