With a 6-3, 6-3, 4-6, 6-7(7), 7-6(10-5) stunning triumph, Murray moved into the next round defeating resilient Berrettini in a match that lasted for four hours, 49 minutes to secure the statement result in Rod Laver Arena.
The tie hung by the thread but the Polish pair prevailed over their Italian counterparts in straight sets to claim the victory. They won by 6-1, 6-2 in the crucial match.
Berrettini got the better of World No 3 in a much-anticipated match, winning it 6-4, 6-4 in straight sets to seal Italy's victory over Norway and take a commanding 3-0 lead. The Italian team will face either Poland or Switzerland in the City Finals on Wednesday.
Auger-Aliassime produced a stunning performance to clinch the match 6-7(3), 6-4, 6-2 against Carlos Alcaraz. The 19-year-old Alcaraz fought valiantly but was unable to close out the tie.
The second set was a repeat of the first and was bagged by the Fench in a stunning display of tennis. He did not give anything away, forcing the opponent to make errors and then capitalizing on them.
The Italian No.8 seed, who had won back-to-back grass titles during his Wimbledon preparations, said he was "heartbroken" to have to miss out. "The dream is over for this year but I will come back stronger."