The former lord advocate said she stepped aside after a "mutual decision" taken with Scotland's top civil servant, Sir Peter Housden.
Lord Fraser of Carmyllie, the other member of the panel that usually investigates alleged breaches of ministerial rules, has said he was "stood down" by Sir Peter.
It means the inquiry – into claims the First Minister broke the ministerial code of conduct by revealing the existence of legal advice on an independent Scotland's membership of the EU – will be conducted solely by former civil servant Sir David Bell, who was drafted in specially for the job.
Dame Elish, now principal of St Hugh's College, Oxford, said: "Because it concerns the provision of legal advice, Lord Fraser and I will not be involved in the matter. The First Minister has invited Sir David to join the panel and on this occasion he will conduct the inquiry rather than the two of us."
She said an "absolutely mutual decision" was taken following discussions with Sir Peter, the Permanent Secretary at the Scottish Government.
The inquiry was launched last week after claims the First Minister lied during a TV interview in March in which he appeared to confirm law officers had given specific advice on Scotland's membership of the EU.
The First Minister denied the claims, insisting he was speaking about Government documents which were "underpinned" by general legal advice.
However, he agreed to an inquiry into claims he broke the ministerial code, which prohibits revealing the existence of any legal advice without permission from the Lord Advocate.
At the time, he told MSPs that Sir David, a former Whitehall colleague of Sir Peter, would join the usual panel of investigators, Dame Elish and Lord Fraser.
He said Sir David would lead the probe but added: "The findings of the independent advisers will be made public."
Lewis Macdonald, Scottish Labour's justice spokesman, said: "The First Minister cannot even manage to be straight with people when it he is appointing advisers to investigate himself.
"Alex Salmond clearly stated David Bell would be joining a panel of advisers to look into this alleged breach. Now it appears he is the sole investigator.
"Why did the Scottish Government make no attempt to clarify the position of Elish Angiolini and Lord Fraser? It is becoming increasingly difficult to have any faith in this bungled process where Alex Salmond writes the law, hires and fires the judges and even decides whether or not he gets charged."
The First Minister's chief political spokesman said: "We have said the inquiry will be led by Sir David Bell. The other two members remain on the panel.
"They will be available to be consulted if that is deemed appropriate. This is no different from previous ministerial code inquiries which have been led by one panel member."
MSPs will today debate the ongoing EU row.
The Holyrood clash comes a week after Deputy First Minister Nicola Sturgeon revealed ministers were only now seeking specific advice on EU membership from their law officers following the Edinburgh Agreement on the referendum.
Labour MSP Paul Martin has called on Mr Salmond to repay £12,000 of public money spent on court action aimed at keeping the existence of legal advice secret.