Ospreys had only one Pro14 victory to its name in the competition against Benetton Treviso back in October, but doubled that tally with a shock success as Storm Dennis battered Swansea.
Ulster, second in Conference A behind Leinster, came from 23-10 down to take a one-point lead courtesy of tries from Matt Faddes and Stuart McCloskey, Bill Johnston converting both scores in the space of nine minutes.
Scott Otten had scored a try in each half for the home side after Owen Watkin's five-pointer, but it looked set for more agony until Price split the posts with his third penalty of the match five minutes from time to lift the gloom.
The Leinster juggernaut rolled on at RDS Arena, where Will Connors grabbed a double and there were also tries for Michael Bent, Ronan Kelleher and Rhys Ruddock in a commanding 36-12 victory.
Luan de Bruin and Aidon Davis crossed for Cheetahs, third in Conference A, but it was soundly beaten by the defending champion.
Edinburgh completed a double over the Scarlets with a 14-9 win in Llanelli to return to the summit of Conference B, Duhan van der Merwe and Matt Scott scoring the only tries.
Connacht thumped Cardiff Blues 29-0, while the encounter between Dragons and Treviso, due to be played at Rodney Parade, was called off because of extreme weather conditions.