Vaughan will fly home on Saturday for treatment, they said, adding the influential skipper was expected to be fit for next month's World Cup.
"Michael Vaughan's hamstring injury will require one week to 10 days recovery period," said a brief statement released by the England and Wales Cricket Board.
"He's expected to make a full recovery from the World Cup."
Vaughan expressed disappointment at being ruled out of the best-of-three finals with Australia but said he was confident of playing in the World Cup.
"It has been very disappointing on this tour to have no problems with my right knee and then a hamstring comes about," he said here on Thursday.
"I am desperate to play, but we have to look at the future and certainly the next few weeks with the World Cup in mind.
"The medical staff have told me there is absolutely no doubt I will be on the plane for the World Cup."
Vaughan's departure will again see all-rounder Andrew Flintoff thrust into the captaincy role for the tri-series finals.
Vaughan missed England's 0-5 Ashes Test drubbing as he continued his recovery from knee surgery, but then suffered the left hamstring tear when he returned for the tri-series.
The problem resurfaced during Tuesday's crucial win over New Zealand, forcing him out of the finals which start here on Friday.
It is yet another setback for the 32-year-old, who helped England to two wins in the three tri-series matches he played after his extended stint on the sidelines.
Vaughan looked in good touch with the bat in the first two games without making a big score, but strained his hamstring in Hobart on January 16.
It was initially assessed as a slight grade one tear, but he aggravated it again in training in Perth and missed five matches before returning against the Black Caps on Tuesday.
Although he made a duck, his astute leadership was evident in the field as England strangled New Zealand's run chase to win by 14 runs.
Vaughan becomes the fifth England player to fly home injured after Kevin Pietersen, Chris Tremlett, Jon Lewis and James Anderson.
Opening batsman Marcus Trescothick left before the Ashes series with a stress-related illness, missing a generally forgettable tour.
England's first World Cup match is just five weeks away, with their opening clash against New Zealand on March 16.