Patrick 'Bonner' Maher scored the crucial goal, as the Premier County beat the Dubs by 1-19 to 0-19 at Semple Stadium.
Tipperary manager Eamon O' Shea told Tipp FM afterwards that he was happy with the performance and his players.
Meanwhile, Dublin have been consigned to a relegation playoff.
Anthony Daly's side will face Waterford in that playoff, after the Deise fell to a 4-22 to 0-14 defeat to Kilkenny at Nowlan Park.
Kilkenny are into the quarter finals after that win.
And Galway and Clare also both make the quarter finals, after playing out a draw in Ennis. It finished Clare 0-18 Galway 1-15.
In Division 1B, Cork have been promoted to the top division, after they beat Wexford at Pairc Ui Rinn by 2-24 to 1-20.
Elsewhere, Limerick beat Laois by 1-17 to 1-13 at O'Moore Park in Portlaoise.
And Offaly thrashed Antrim by 5-27 to 4-09 in Tullamore.
All of that means the quarter final line up sees Clare take on Laois, Kilkenny play Wexford, Galway battle it out with Limerick and Cork play Tipperary.
Tipperary and Cork will toss a coin tomorrow to decide on the venue for their game.
Meanwhile, the relegation play-offs will see Dublin take on Waterford and Antrim play Offaly.
Elsewhere; Tipperary Senior Camogie team have been beaten by Wexford on a 1-16 to 1-06 scoreline in their National League Division 1 game at The Ragg.
But Tipperary defeated Cork in the Munster U16 A Camogie Final by 1-10 to 0-10 in Limerick.
