The minister, David Choquehuanca, told reporters France and Portugal had closed their airspace to the presidential plane over the "huge lie" that Snowden was on board. Defense Minister Ruben Saavedra said Italy closed its airspace as well.
Morales had been on a visit to Moscow, where Snowden -- who is facing espionage charges in the United States -- has been holed up in an airport transit area for more than a week.
"The president was forced to land in Vienna," said Choquehuanca, alleging that Morales's life had been put in danger by what he called a forced emergency landing.
"There were unfounded rumors that Mr Snowden may have been on board the aircraft. We have no idea who made up this huge lie."
The Bolivian leader was attending a meeting of natural gas-producing nations in Moscow and had met with Russian President Vladimir Putin.
Earlier, Morales told Russian media his country would "debate and consider" granting political asylum to Snowden "if there were a request."
Bolivia is one of 21 nations to which Snowden had applied for asylum, according to the anti-secrecy WikiLeaks website.
"Bolivia is there to shield the (accused), whether it is espionage or control. In either case, we are here to assist," Morales told Russia's state-run RT television in comments translated by the channel from Spanish.
Choquehuanca said Morales's plane had been scheduled to refuel in Portugal, but both Lisbon and then Paris denied overflight.
"Inexplicably we were informed (by Lisbon) that the overflight and landing permission had been cancelled," the top diplomat said.
The plane was re-routed to include a stop in Spain's Canary islands, but France then refused to allow overflight of its territory, Choquehuanca said.
"We would like to make it known that we are unhappy and upset because the president's life has been put in danger," he said.
Saavedra told Telesur news network that just an hour before entering France's airspace, the plane was told not to enter, with authorities citing unspecified technical reasons.
"They violated air traffic rights," a visibly upset Choquehuanca said, maintaining Morales' life had been in jeopardy.
"They are trying to intimidate us. This is discrimination," Choquehuanca argued.
France's foreign ministry said it could not immediately comment on the situation.
Some hours later, Saavedra told state media from Austria that France now had approved overflight by Morales's plane. An Austrian foreign ministry official said Morales would leave "early Wednesday" for La Paz.
Bolivia's leftist regional allies Ecuador, Nicaragua and Venezuela rallied behind Morales, voicing outrage.
"This is an attack against President Morales's life," Venezuelan Foreign Minister Elias Jaua said. "We must make ourselves heard to the European and US governments who are behind this outlandish and brutal, uncivilized attack, putting a president's life in danger."
Snowden, 30, revealed details of a vast US surveillance program to collect phone and Internet data. His passport was revoked after US federal charges were filed against him.