Postecoglou clinched the Manager of the Month awards for August and September after enjoying a fine start to life in the Premier League.
Having been named the winner of the October prize, the Australian is now the first coach to begin his career in England with three consecutive Manager of the Month awards.
The 58-year-old has also become the fourth manager to win the award for a third successive month, following in the footsteps of Antonio Conte, Pep Guardiola and Jurgen Klopp.
Postecoglou earned the recognition after seeing his side claim three straight victories in October, beginning with a narrow 1-0 win over newly-promoted Luton Town.
They went on to record a 2-0 home victory against Fulham, before a Joel Ward own goal and a Son Heung-min effort guided them to a 2-1 win in their away meeting with Crystal Palace.
Spurs ultimately finished October in top spot after winning eight and drawing two of their first 10 Premier League matches.
However, their unbeaten start to the season was brought to an end on Monday when they were beaten 4-1 by Chelsea in a game where both Cristian Romero and Destiny Udogie were shown red cards.
Meanwhile, Salah has become the first Liverpool player to win the monthly award since Joel Matip, who earned the prize in February 2022.
The Egyptian, who is now a five-time winner of the award, has been named Premier League Player of the Month for the first time since October 2022.
Salah enjoyed a superb spell of form in October, scoring five goals in three matches to help Liverpool collect seven points.
Impressively, the 31-year-old netted all four of Liverpool's goals across a 2-2 draw with Brighton & Hove Albion and a 2-0 victory over Everton.
The Liverpool forward rounded off the month with a strike in a comfortable 3-0 victory over Nottingham Forest, taking his Premier League tally to eight goals for the season.
After failing to trouble the scorers in last weekend's 1-1 draw against Luton, Salah will hope to find the net when the Reds face Brentford on Sunday.