The Ospreys, who outscored their opponents by three tries to two through Jones, Lee Byrne and Nikki Walker, then decided to go for a long-range Biggar drop goal with time against them.
But the fly-half, who missed five kicks at goal in the Basque country, saw his effort fall short as Biarritz surged into the semi-finals for the first time since 2006.
In the semi they will have home-country advantage over Munster, who defeated the French side in the 2006 final.
"We are really happy about the result," said Biarritz captain Jerome Thion after the Estadio Anoeta victory.
"The Ospreys were very intense and reactive and it was a close game. But we deserved the victory and look forward to the semi-finals."
Ospreys coach Sean Holley shared Jones' uneasy feelings about the game's denouement, though.
"I will certainly ask him [Clancy, about the late decision]," said Holley.
"It's a tough call that has to be made at the end but I have had another look at the incident and it looks like a penalty offence to me."
It is the third successive season that the Ospreys have exited Europe's premier competition at the last-eight stage.
But they will regret this opportunity after spurning a number of gilt-edged chances during the compelling tie.
"We have to start believing and realising the potential of the side," said Holley.
"We have great players but we are a bloody good team. We're not a team of individuals.
"We made a lot of opportunities in the first half, James Hook was outstanding and Mike Phillips really stepped up to the plate.
"We made a lot of breaks with some great attacking play and showed plenty of endeavour but we just lacked a clinical edge.
"It's extremely frustrating. The players do not need to be told that we need to start nailing those chances."
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