The diplomatic spat began after the match, when Venezuelan players accused Peruvian police of beating them when they went to greet fans on Tuesday night. It was a tumultuous night for the South American qualifiers, after a match between Brazil and Argentina in Rio de Janeiro led to ugly clashes between fans.
"The Peruvian government has carried out another arbitrary act against the Venezuelans by preventing the plane bringing the team back from refuelling," Foreign Minister Yvan Gil wrote on social media.
"This is kidnapping as vengeance against our team which played an extraordinary match" in Lima, he added.
The Peruvian government said that it "has not ordered any measure that prohibits the refuelling" of the aircraft" and that "it is taking steps to resolve this situation as soon as possible."
"The aforementioned aircraft has been experiencing private commercial supply restrictions beyond the control of the Peruvian State," he added. Hours earlier, Venezuelan footballer Nahuel Ferraresi showed bandages on his right hand, saying Peruvian police "beat me".
"These are things which should not happen. The match was over and we went to thank our Venezuelan supporters," he said.
Ferraresi said another player had handed his jersey over to fans, and he was about to throw his to the crowd when the police blocked him. "Then others got angry, I don't know what happened, and (the police) took out their batons to hit us. They hit me twice... but it's not a serious injury."
Videos of the incident, which show Peruvian police officers brandishing their batons against the players, have gone viral on social media. Venezuela's football federation condemned "acts of discrimination and xenophobia" against the team and their fans.
After the draw, Peru is in last place in the qualifiers, while Venezuela - which has never qualified for a World Cup - is in an unprecedented fourth place. Before the match, Peruvian police carried out an unusual identity check among fans, and were accused of targeting Venezuelans.
This move comes a week after a controversial decree came into force authorizing the government to expel foreigners. Peru is home to some 1.5 million Venezuelan immigrants, many of whom have fled a crippling economic crisis at home.