Michael Kightly scored the only goal of the game in the 65th minute to seal Burnley's first win in six league matches, enough to send them above Leicester and QPR, who both dropped points.
Leicester, who had been top at the start of the day, lost 3-1 at Brighton and Hove Albion while QPR were held 0-0 by Blackburn Rovers at Loftus Road.
Andy King pulled one back for Nigel Pearson's side but Barnes made sure of the result with a penalty 13 minutes from time as Leicester lost their second successive game.
Martin completed his hat-trick with another penalty 12 minutes later after Neal Bishop had been sent off, before Angel Martinez became Blackpool's fifth red card in two matches, Richard Keogh completing a miserable afternoon for the visitors by rounding off the scoring in the 90th minute.
Three points behind Derby in fifth are local rivals Nottingham Forest, who won 1-0 at Sheffield Wednesday thanks to Simon Cox's 48th-minute winner.
Goals from Lewis Grabban and Matt Ritchie shortly before half-time put the visitors in command, with Adam Le Fondre's 90th-minute strike proving nothing more than consolation.
At the wrong end of the table, Yeovil Town and Charlton Athletic finished 2-2 in a result which does little for either side's hopes of pulling away from the bottom three.
