The Lothians MSP said she wants people who voted Yes to Scotland leaving the UK last year to feel they have “a home” in the Labour party.
The surprise move represents a major shift for the party that was at the heart of the pro-union Better Together campaign during the referendum.
Although the vote was won by the No side, around one in three Labour voters backed independence and have since abandoned the party.
Speculation is now growing over the prospect of a second independence referendum, with recent polls suggesting Scotland is split 50-50 on the issue.
Ms Dugdale’s intervention came after former leader Johann Lamont suggested the party should have a free vote in any future vote on leaving the UK. Ms Dugdale, who replaced Jim Murphy as Labour leader in Scotland last month, said she sympathised with Ms Lamont’s position.
She said: “I would like to think that people who supported Yes in the referendum and who might have that view again in the future have a home in the Scottish Labour party.
“I want people who voted both Yes and No to see that the Labour Party is the vehicle for progressive change in this country, which is why I am completely comfortable and, in fact, would encourage people who have voted Yes in the past to take a look at our party and see that it is changing.”
Asked whether Labour MPs or MSPs should be able to argue for independence, she told the BBC’s Sunday Politics Scotland programme: “Yes. Many Labour party members, in fact almost 30 per cent of Labour party supporters, voted Yes. We know that now from all the evidence and I respect that. I’m not going to shut down my party’s renewal and debate because people hold a different position around independence.
“We should have a democratic debate within our party over the big issues of the day.”
First Minister Nicola Sturgeon warned last week that the SNP’s manifesto for next May’s Holyrood election will include the provision for a second referendum and the potential “triggers” for this.
She said in a speech on the anniversary of the referendum on Friday that the UK is living on “borrowed time.”
Ms Dugdale’s announcement was welcomed as “progress” by senior Nationalist Pete Wishart MP, who chairs the Scottish affairs select committee.
He said Labour is “beginning to understand that they can’t face down their ‘former’ voters” on the independence issue.
“My prediction is that SLab (Scottish Labour) will become a party of independence following an even worse, catastrophic result next year,” he told followers on social media. “Then Scotland wins.”
Labour’s association with the Conservatives on the Better Together platform was widely seen to have damaged the party in the aftermath of the referendum and contributed to its rout in this year’s general election, in which it lost 40 of its 41 seats.
With polls suggesting the nationalists are on course for another Holyrood landslide in next May’s Scottish Parliament vote, the new Labour leader may be attempting to reach out to Yes voters.
But her move was met with a withering response from Scottish Conservative leader Ruth Davidson.
Ms Davidson said: “For the leader of a party supposedly committed to safeguarding Scotland’s place within the United Kingdom, it beggars belief that Kezia Dugdale should now be giving her support to Labour MSPs and activists who want to campaign for independence.
“As a result of her statement, the position of the Scottish Labour party on this most vital of issues has been plunged into complete confusion and Kezia Dugdale must explain her apparent U-turn immediately.
“This further lack of clarity from Scottish Labour makes it clear that the Scottish Conservatives are the only party who are both unequivocal in their commitment to securing Scotland’s place within the United Kingdom and who have the strength and resolve to stand up to the SNP and its continuing drive for independence.”
Ms Lamont said that the party should debate whether to have a free vote in any future independence referendum.
She told BBC Radio Scotland’s Good Morning Scotland programme: “We know that there were Labour people who supported Yes and we need to understand what that was and have that debate inside the party.
“It is for others to decide whether you then make it a free vote. People clearly exercised their vote in the ballot box anyway.
“If that’s a debate we should have inside the party, why not?”
Ms Lamont said that “on balance” she believed that joining forces with the Conservatives in the Better Together campaign had been the right thing to do, despite Labour’s subsequent disastrous defeat in May’s General Election.
Ms Dugdale has been clear that she and her deputy Alex Rowley will be in charge of Labour policies in Scotland after Ms Lamont quit claiming the party was being treated like a “branch office” under former leader Ed Miliband.
She said yesterday: “I’m the leader of the Scottish Labour party and the positions we’re going to take will be set here in Scotland by me - I think that’s what people in Scotland expect.”
Ms Dugdale has said she does not share new leader Jeremy Corbyn’s anti-Trident position, but takes the “multi-lateralist” view that the UK should only scrap its nuclear deterrent when other nuclear powers do the same. Ms Dugdale also distanced herself from Mr Corbyn’s plans to renationalise the energy companies insisting this was “not the priority” for the moment.
Senior Labour politicians have previously stood on opposite sides of the constitutional debate.
Brian Wilson, who went on to become a Labour energy minister under Tony Blair, campaigned against the party’s plans for Scottish devolution in the late 1970s. A few years earlier the party had also been split over the referendum on entering the EEC with senior figures such as Tony Benn and Michael Foot opposed to leader and Prime Minister Harold Wilson’s support for joining.
The latest poll of polls published by Professor John Curtice of Strathclyde University last week suggested the SNP is heading for a second overall majority in next May’s Holyrood vote. The nationalists have an average of 54 per cent, with Labour trailing on 22 per cent.
Professor Curtice said this was down to those backing independence last September – who included one in three of those who voted Labour in 2010 – wishing to “affirm” their choice by voting SNP.
This has given Ms Sturgeon’s party a foundation of support that “none of their opponents can match”.
About 90 per cent of those who voted Yes a year ago went on to back the SNP in May’s Westminster election rout north of the Border. That pattern, together with some success in picking up No voters, was enough to take the SNP to the 50 per cent mark.