But Stade substitute Lionel Beauxis booted a 50-metre penalty in the fifth minute of injury-time, meaning the Parisians require a bonus-point victory against Italian minnows Calvisano on Saturday for a prized last-eight place.
The Ospreys, meanwhile, visit English champions Sale Sharks next weekend, when a bonus-point success could still be enough to clinch one of two best runners-up spots in the knockout stages.
Although television footage suggested Ospreys flanker Steve Tandy had been penalised for offside by English referee Chris White during Sunday's last-gasp penalty incident, number eight Jones admitted he was guilty of a hand in the ruck.
Jones said: "It was me who gave away the penalty. I didn't think he [Beauxis] would make it from that range, but he did, and I had to say sorry in the dressing room afterwards.
"Stade are seasoned performers in the Heineken Cup, and they did a real job on us in the first-half, but we regrouped in the second period.
"We are pretty disappointed - we didn't deserve to win the way we played in the first-half."
Stade led 19-6 after 60 minutes through a Juan Martin Hernandez try, 11 points from fly-half David Skrela and a Beauxis penalty, before Ospreys stirred and went ahead after James Hook kicked five penalties and converted wing Nikki Walker's touchdown.
Ospreys coach Lyn Jones said: "It was quite a ride of emotions out there, but we are very disappointed with the outcome. We gave Stade a lot of points.
"We lost flanker Filo Tiatia with a torn calf muscle after just three minutes, and perhaps it just wasn't going to be our day."
As for the daunting Sale mission, Jones added: "In the past, 21 points has often been good enough to reach the quarter-finals, and that total is the best we can look for now.
"Going to Sale and winning will be difficult enough, let alone scoring four tries. I am a realist and I know how difficult it will be next weekend."
Stade, meanwhile, were left breathing a huge collective sigh of relief after securing a share of the spoils from a game they looked to have wrapped up with only 20 minutes remaining.
Scrum-half Agustin Pichot said: "We were very close to throwing the game away. When you think you have won, you can sometimes lose your grip and let it go, and we made some mistakes.
"We performed really well in the first-half - we didn't let them (Ospreys) play. We had control of the game, but we always seem to lose a bit of concentration away from home.
"For many of us at Stade Francais, this season is probably our last chance of winning the Heineken Cup, and that is what our goal is."