Another Messi double brings his annual total to a staggering 88
But last year's runners-up Athletic Bilbao were dumped out by third-tier Eibar on away goals.
On Sunday, World Player of the Year Messi overhauled Gerd Mueller's best for a calendar year of 85 set in 1972, and he opened the scoring in the 11th minute at Cordoba's Arcangel stadium when he clipped the ball into the net off the underside of the crossbar from close in from a David Villa centre.
Fielding a near full-strength team despite the lowly opposition, Barca controlled the game in the second half before Thiago Alcantara sent Messi clear on the left in the 74th minute and he finished low into the corner.
Bilbao's disappointing season took a turn for the worse when Mikel Arruabarrena's 72nd-minute penalty for Eibar at the San Mames knocked their Basque neighbours out after the second leg of their postponed last-32 tie ended 1-1.
Eibar, who held Bilbao to a 0-0 draw in the first leg, went through on the away goals rule and will play Malaga in the last 16, with a quarter-final against Barca or Cordoba the prize for the winners.
On the other side of the draw, Sevilla virtually assured themselves of a berth in the last eight when they crushed La Liga rivals Real Mallorca 5-0 in Palma, Alvaro Negredo scoring twice to set the 2010 winners on their way.
Valencia moved closer to a place in the quarter-finals and a possible meeting with Real when Dani Parejo and substitute Roberto Soldado struck in a 2-0 win at Osasuna in their last-16 first leg on Tuesday.