The crew, along with a number of tourists, were drawn to Etna in Sicily to observe the spectacle of the active volcano erupting but were caught by surprise when flowing magma hit thick snow, causing an explosion that caused rock and other material to rain down upon them.
Volcanic rocks and steam from the eruption injured at least 10 people, witnesses and media reported, and the network's science reporter who was on assignment on Etna, Rebecca Morelle, described the experience in a series of tweets.
"Running down a mountain pelted by rocks, dodging burning boulders and boiling steam — not an experience I ever ever want to repeat," she wrote.
The BBC crew was shaken but mostly unharmed, having suffered minor cuts, bruises and burns, she wrote.
Morelle said the explosion was "a reminder of how dangerous [and] unpredictable volcanoes can be".
Etna active for past two days
Authorities said about 35 tourists were in an area they had permission to enter on the volcano, when the explosion occurred about midday on Thursday (local time).
The guides who accompanied them helped bring them to safety.
The president of the Italian Alpine Club chapter in Catania, Umberto Marino, was traveling up the volcano in a snowcat when injured people started running in his direction.
"The material thrown into the air fell back down, striking the heads and bodies of people who were closest," Mr Marino was quoted by local media as saying.
According to Italian news agency ANSA, four people including three German tourists were hospitalised, mostly with head injuries — though none of the injuries was listed as grave.
Among those present when the explosion occurred was a scientist from Italy's volcanology institute, Boris Behncke, who wrote on his Facebook page that he had suffered a bruise to his head.
"I am generally fine and having a good, well-deserved beer in this moment," Mr Behncke said.
Mount Etna has been active for the past two days, creating a visual spectacle as it spews lava and ash into the air.
A new lava flow started from the south-eastern crater on Wednesday and was advancing with a temperature above 1000 degrees Celsius in an area covered by snow.
Officials at Sicily's Catania airport announced they would reduce arrivals by half to five flights an hour, due to ash clouds.
Departures would continue as scheduled.
Italy's volcanology institute said it was continuing to monitor the phenomenon.