Kountouris, who was with the Sri Lankan team from 1996-2003, has worked for the past three years under Alcott, who has held the job since 1984.
Kountouris has also stood in for Alcott during the tours of South Africa and Bangladesh earlier this year.
''It goes without saying that Errol has played a significant role in the success of Australian Test and one-day international cricket over the past 22 years,'' Cricket Australia's General Manager Cricket Operations Michael Brown said in a statement today.
''Alex was the Sri Lanka physio for six years before becoming Errol's understudy and Alex handled a collection of demanding situations and injuries on the recent tours of South Africa and Bangladesh,'' he added.
Kountouris said he was delighted to have won the nod, and Alcott has also endorsed the decision.
''It is a fantastic honour to be given this responsibility and a really exciting challenge,'' Kountouris said.
Kountouris has also helped develop and co-ordinate the Master of Sports Physiotherapy course at Latrobe University in Melbourne and is currently conducting Cricket Australia-funded research into fast bowling injuries, having worked in football before becoming a cricket physio.