The celebrated US fashion designer was found dead in her New York flat on Monday in an apparent suicide.
Sir Mick earlier said he failed "to understand how my lover and best friend could end her life in this tragic way".
In a statement on their website, the band thanked fans for their support.
They said: "The Rolling Stones are deeply sorry and disappointed to announce the postponement of the rest of their 14 On Fire tour of Australia and New Zealand following the death of L'Wren Scott.
"Mick Jagger, Keith Richards, Charlie Watts and Ronnie Wood wish to thank all of their fans for their support at this difficult time and hope that they will fully understand the reason for this announcement.
"The Rolling Stones are planning to reschedule these postponed shows at a later date."
The British rock band had been due to begin the tour in Perth on Wednesday, having flown in to the Western Australian city on Sunday.
On his website, Sir Mick, who started dating Scott in 2001, said they had spent "many wonderful years together".
He described his partner as having "had great presence" and said "her talent was much admired, not least by me".
Sir Mick added: "I have been touched by the tributes that people have paid to her, and also the personal messages of support that I have received. I will never forget her."
Scott, 49, was found by her assistant at 10:00 local time (14:00 GMT) on Monday.
Ninety minutes earlier, Scott had sent her assistant a text message asking her to come to her Manhattan apartment, without specifying the reason why, the Associated Press news agency reported.
Police said there was no sign of foul play and no note has been found.
Supermodel Naomi Campbell, Vogue editor Anna Wintour and singer Madonna were among those who were fans of the designer.
The former model founded her own high-end fashion label in 2006 and created a more affordable line of clothes with Banana Republic. She was also a costume consultant on films including Ocean's Thirteen and Eyes Wide Shut.
Wintour described Scott as "a total perfectionist... always unbelievably generous, gracious, kind and so much fun."
Madonna wrote in a statement: "This is a horrible and tragic loss. I'm so upset. I loved L'Wren's work and she was always so generous with me."
Actress Nicole Kidman, said to be a friend of many years, was "heartbroken and in shock right now and unable to say anything", according to a spokeswoman.
Bianca Jagger, one of Sir Mick's former wives, tweeted: "Heartbroken to learn of the loss of the lovely and talented L'Wren Scott. My thoughts and prayers are with her family. May she rest in peace."
The UK's Press Complaints Commission (PCC) told the BBC it had received a small number of complaints from the public over images published which apparently showed an upset Sir Mick at the time he was told Scott had died.
It is understood the singer has not yet complained personally, but the PCC said it would deal with the matter in due course.
After Perth the Rolling Stones had planned to perform concerts in Adelaide, Sydney, Melbourne, Macedon and Brisbane before flying to Auckland, New Zealand, for the final show on 5 April.
Ticket holders for the affected shows have been advised to keep hold of their tickets and await further information.
The On Fire tour is scheduled to move to Europe in June, with concerts planned for the Netherlands, Germany, Italy and Belgium.

Local photographer Ron Strathdee captured the phenomenon on Monday at about 23:30 BST.
The glow is usually best seen from northern latitudes like Norway, Alaska, Iceland and northern Scotland.
Mr Strathdee said seeing the Northrn Lights from Manx latitudes was "fairly unusual."
They happen when incoming solar radiation hits the earth's upper atmosphere and excites atoms to a new energy state, emitting energy in the form of light.
The photographer said: "I needed a place that faced north so went to Peel Hill and tried some shots over the castle which worked but half the fishing boats in the Irish Sea were discharging fish at the breakwater with enough floodlights to cover a football match!
"Going round the front of the castle it was pitch dark and it looks straight north which was where the photos were taken."
The Rolling Stones have postponed their Australia and New Zealand tour after the death of Sir Mick Jagger's girlfriend L'Wren Scott.
[NEXT_CONCEPT]
The Aurora Borealis, better known as the Northern Lights, has been photographed from Peel Castle on the west coast of the Isle of Man.