Stamford Bridge was braced for a draw before the Spain international popped up to add to an earlier strike from Eden Hazard and secure maximum points.
The Chilean opened the scoring by firing home a rebound before Muller made the result safe after enduring a barren Euro 2016 campaign for his country.
Jurgen Klopp's side returned to winning ways in some style with Divock Origi also on target as the hosts lose again to increase their fears of relegation.
Despite being frustrated for large spells of the match, the Catalans found their breakthrough in the second half to go within three points of La Liga leaders Real Madrid.
The United States international stole the headlines on home soil as he inspired Carlo Ancelotti's men to an impressive victory over the Serie A side.