Barnes was judged to have been brought down by goalkeeper Alisson Becker, picked himself up, dusted himself off and scored the only goal of the game.
Divock Origi, starting in place of the dropped Roberto Firmino, had earlier hit the crossbar, before Mohamed Salah, another substitute, was denied twice by Nick Pope.
It was Burnley’s first at Anfield for 46 years and ended Liverpool’s 68-game unbeaten run, leaving Chelsea’s superior record intact.
Chelsea’s 86-game unbeaten record at Stamford Bridge remains the longest home unbeaten run in Premier League history.
The Blues were unbeaten at home in the top flight between February 2004 and October 2008, with Liverpool the team ending that run.