Turf Moor is generally a tough place to score, but there were no such problems for Rodrigo, who struck twice in two minutes after coming off the bench.
It came after Mateusz Klich opened the scoring with a fine first-half strike and a deflection off Jack Harrison opened the floodgates after the break.
Speaking on commentary for talkSPORT, former Manchester City and West Ham winger Trevor Sinclair described it as a fantastic performance from the visitors.
He said: "Second-half, they were exciting, dynamic, clinical and brilliant. They were absolutely superb.
"The goals they scored were so easy on the eye, such great team goals, one and two touch, getting players into the box. It was an array of different goals.
"They were exceptional from bottom to top."
Neither team had anything to play for, with the Clarets safe and the Whites all-but out of contention for Europe.
However, Bielsa's men would love to finish inside the top ten after all the pre-season doubts following their promotion fro the Championship.
They delivered another display which proved they belong among the country's top clubs, becoming the first to hit four past Burnley at Turf Moor since 2019.