Safarova, the Czech world number 29, was expected to make short work of a player ranked 161 places lower in the world, but instead slumped to a 6-1 6-0 defeat.
On a day of surprises at the Romai Tennis Academy, second seed Alize Cornet of France also made a swift exit as she lost 6-4 6-2 to Andrea Hlavackova of the Czech Republic.
Sixth seed Chanelle Scheepers of South Africa is safely through to the second round after seeing off Slovakia's Michaela Honcova 6-4 6-1, while eighth seed Anna Tatishvili of Georgia was also a winner, ending the hopes of Reka-Luca Jani 6-3 6-4.
Elsewhere on Tuesday, there were wins for Petra Martic, Shahar Peer, Yvonne Meusburger, Alexandra Cadantu, Danka Kovinic and Timea Babos.
Britain's Johanna Konta crashed out in the first round of the Budapest Grand Prix.
The British number three rallied from a set down to take her match with Montenegro's Danka Kovinic, a player ranked 24 places below her in the world, to three sets.
But the 22-year-old ran out of steam in the decider, eventually succumbing to a 6-2 3-6 6-2 loss in one hour 53 minutes.