Rob Street and Brandon Aveiro struck for Palace in the first half before Ali Koiki pulled one back for the home side on 67 minutes with his ninth goal of the season.
Legzdins was the unlikeliest of heroes as he headed in from a corner to keep Burnley second.
Kareem Isiaka netted a second-half winner as Charlton Athletic won their first PL Cup match of the season, beating bottom-club Colchester United 1-0.
Lucas Boye scored the winner on his return from injury to send Reading top of the table with four points.
Chris Willock set up the first-half goal that keeps West Brom top, on six points, three more than second-placed Newcastle.
The Cherries twice had to come behind at Vitality Stadium after goals from Stoke's Daniel Jarvis and Scott Wara.
Alex Dobre scored the first equaliser before, with 11 minutes left, Anthony converted a spot-kick that he had won himself to keep the hosts two points ahead of the Potters.