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The 25-year-old World Player of the Year earlier equalled Mueller's mark when he picked up the ball in a central position in the 16th minute and left the Betis defence standing before sending an angled shot across goal into the bottom right corner.
Andres Iniesta's clever back heel set the Argentine up to score again nine minutes later with a sweeping shot that flew into the same corner of the goal.
Favourite to secure a fourth successive World Player award when the winner is announced next month, Messi has netted 74 times for Barca and 12 for Argentina in 2012, beating Mueller's total for Bayern Munich and Germany in 1972.
He has scored 56 in La Liga, 13 in the Champions League, three in the King's Cup and two in the Spanish Super Cup.
Mueller, known as the "torpedo" while Messi's nickname is "the flea", netted 72 for Bayern and 13 for Germany in 1972 and was 27 when he set the record.
Messi's double took his tally in 15 La Liga outings this season to 23 and he also became Barca's all-time top league scorer on 191 goals, one more than Cesar Rodriguez.