The three families, including six children, are all Muslim and had their homes bombed during the Syrian war.
The Vatican said in a statement that Pope Francis wanted to "make a gesture of welcome'' to the refugees.
Thousands of migrants are now stuck on Lesbos after last month's EU-Turkey deal to try to ease the flow.
All of those leaving with the Pope were already living on Lesbos before the deal was implemented, the Vatican said.
Under the EU-Turkey agreement, migrants arriving illegally on the Greek islands from Turkey after 20 March will be deported unless they successfully claim for asylum.
In return, for every Syrian returned to Turkey, the EU will take another Syrian directly from Turkey.
Pope Francis earlier told migrants living in the Moria camp - some of whom are facing being sent back - "you are not alone".