Having won all four of the previous meetings between the two, Cirstea may have been confident of booking a place in the quarter-finals with another victory against her Swiss opponent.
However, Vogele produced an inspired performance to destroy the Romanian in just 49 minutes, surging to an emphatic 6-1 6-1 success on the hard court in Azerbaijan.
After breaking Vogele immediately, Cirstea won just one service game in the match as the world number 72 set up a meeting with Shahar Peer of Israel.
Second seed Elina Svitolina experienced no such problems, overcoming Spain's Silvia Soler-Espinosa 7-5 6-2.
Ukrainian Svitolina broke Soler-Espinosa's serve four times in a straighforward victory, earning a quarter-final clash with France's Pauline Parmentier.
Meanwhie, Kristina Mladenovic and Francesca Schiavone will do battle in the last eight after both kept hopes of winning the trophy the WTA international tournament alive with victories on Thursday.
Schiavone staved off the threat of a fightback from Julia Glushko to defeat the Israeli 6-2 7-6 (7-2).
Mladenovic was made to work extremely hard for her place in the last eight, coming from behind to beat promising Croatian teenager Donna Vekic 2-6 6-3 6-0.
Vekic, who won her first WTA title at the Malaysian Open back in April, looked to be on her way to victory after taking the first set in comfortable fashion.
However, an error-strewn showing, in which she totalled 10 double faults and won just six points on her second serve, allowed France's Mladenovic to battle back and progress.