Tipperary overcame Galway in the best hurling game so far this year by 3-17 to 3-16 in a thrilling match at Croke Park.
Galway led by two points with minutes remaining but Tipp staged a storming comeback to win at the wire.
They will now face Waterford in the hurling all Ireland semi final.
In the other qualifier, Cork easily defeated rank outsiders Antrim by 1-25 to 0-19 and will now play Kilkenny in the other All Ireland semi final.
Three million people in the world are descended from one Irish High King