The party, led in Scotland by Glasgow MSP Johann Lamont, is calling on new Scottish Health Secretary Alex Neil to bring forward laws to change organ donation legislation to move from the current opt-in system to an opt-out.
It came weeks after we revealed that more than 10,000 Evening Times readers had signed the petition to persuade the Scottish Government to switch to a system of presumed consent.
MSP Jackie Baillie, Scottish Labour's shadow health secretary, said the party was "disappointed" there were no plans by the Scottish Government to bring the proposal forward.
She said: "Nicola Sturgeon (former Scottish Health Secretary) failed to agree to change the legislation that is widely thought would save many more lives and bring hope to the thousands of Scots who are waiting for organs on the transplant list.
"Alex Neil has previously stated his support for such a change, which has long been supported by patients groups and the medical community. We hope he will now use his new post to make it happen.
"We will support the Government in introducing legislation."
Central Scotland MSP Mark Griffin, 26, said: "The party recognises there is a massive need for organ donors and the benefits organs bring to those who have been waiting so long.
"I would hope to see it come about within the lifetime of the Parliament, before the elections in 2016.
"I am delighted for the Evening Times campaign and am delighted the Labour Party has publicly backed the campaign."
Earlier this year Mr Griffin told the Evening Times he believed his father Francis, a former Labour councillor for Kilsyth and Croy – who died five years ago aged 47 after waiting almost 10 years for a heart transplant – would still be alive if the law was different.
The MSP said: "My dad didn't make it through the operation because his health had deteriorated so much.
"If the opt-out system had been in place with more donors I think he would have got his transplant a lot earlier and would still be alive."
The Labour move has also been welcomed by the British Medical Association. Dr Brian Keighley, chairman of the BMA in Scotland, said: "We welcome Labour's support for an opt-out system for organ donation to be introduced in Scotland."
The news came as the future of Scotland's organ donation laws were being discussed at a one-day conference in Glasgow.
The event, organised by the Scotland Patients' Association, was taking place today at the Hilton Hotel.
Scottish Public Health Minister Michael Matheson will be among the speakers.
HOW YOU CAN HELP
WE need your help to make this happen and save more lives.
Sign our petition to lend your support to an 'opt-out' system for organ donation at: www.eveningtimes.co.uk
There is also a printable version of the petition available on our website.
We already have more than 10,000 signatures but want thousands more. So we need as many of you as possible to distribute it in community centres, churches, mosques, leisure centres and public places.
You can also sign up on Twitter and Facebook via http://bit.ly/etoptin. Don't forget to send the link to friends.
To join the existing national organ donor register, text 'ETDONATE' to 61611.
Texts are charged at the usual standard network rate (any texts sent in reply are free to the recipient).
You can also join the list by calling the NHS Donor Line on 0300 1232323. Lines are open 24 hours a day. Calls are charged at your contracted rate for local calls.
Those who wish to register online can also do so by filling out a form on: www.organdonation scotland.org/ nhs_organ_ donor_ register. html