Francis Saili's 76th-minute try clinched maximum points.
It leaves Blues trailing the Irish province by six points in the race for sixth place with two games to play.
Danny Wilson's team are three points ahead of Ospreys, who they face in Judgement Day IV in Cardiff's Principality Stadium on Saturday.
Ospreys are now nine points off the pace and need a miracle to claim a Champions Cup place.
Blues need to beat their Welsh rivals and then beat Edinburgh in their final match of the season on Saturday, 7 May to stand any chance of claiming a place in Europe's top flight competition next season.
Even then, they will probably need Scarlets to beat Munster in Ireland in their last game.
The West Wales region are on the same number of points as Munster but have a game in hand.
They face Newport Gwent Dragons in Cardiff on Saturday before their trip to Thomond Park.
If results do not go the Blues' way, it will leave Wales with only one team in the Champions Cup next season - the same representation as Italy.
Cardiff Blues face an uphill battle for European Champions Cup qualification after Munster's 27-19 bonus-point Pro12 win over Edinburgh.