Ministers from more than a dozen European and Latin American states began meeting in Montevideo to negotiate a solution to the crisis as the rival leaders clashed in Caracas over allowing humanitarian aid into the impoverished country.
"The task before us is urgent and this urgency comes from the worsening of the situation that risks destabilizing the entire region, and not only the region," Mogherini told ministers at the start of the meeting.
"Millions of people have already left the country, and many more are fleeing during these hours. "Therefore, it is essential to avoid internal violence and external intervention, and to open a path for a political process that leads to early elections," the EU official said.
Dozens of pro-Maduro supporters protested outside the meeting in Montevideo against US intervention in the country.
But all eyes were on the Venezuelan border at the other end of the continent, where the armed forces have blocked an intended route into the country for desperately-needed humanitarian aid, heightening tensions with Washington.
Hundreds of Venezuelan migrants gathered at Cucuta on the Colombian side of the border to see if they could receive some of the aid being stockpiled in nearby warehouses.
Yajaira Gonzalez, 64, said she was begging Maduro to let through the aid - food, medicine and personal hygiene kits - to her desperate compatriots back home.
"It is not like you say - that we are doing OK," said Gonzalez, referring to Maduro. "Mr. President, we are not OK. We are suffering."
Gonzalez said she used to be a Chavista - a supporter of Maduro's predecessor, the late socialist firebrand Hugo Chavez - but not anymore.
Dajelys Lopez crossed from the other side of the border with her baby in a stroller to see if she could find in Cucuta what she cannot get in Venezuela. "Yesterday a friend died because he suffered a seizure and did not have medication to deal with it," Lopez said.
Guaido has appealed to the army to side with the constitution, from which he takes his legitimacy, than with "a dictatorship that does not have an iota of humanity."
US Secretary of State Mike Pompeo said Venezuela's military was deliberately blocking the aid "under Maduro's orders.