Tyrone suffered late heartbreak at Croke Park where a Dean Rock goal robbed them of victory over the Dubs.
The Red Hands appeared set to hold on for a narrow win after battling heroically, defending with discipline and nailing scores on the breakaway.
But they were denied the two points by a 69th minute strike that rescued Jim Gavin's side who had gone into the match as strong favourites.
Rock finished with an individual tally of a goal and six points.
It was Tyrone's second draw of the Division One campaign and they are on four points from four games, one point better than Dublin.
Sean Cavanagh and Dean Rock swapped early points from frees, before Tyrone defender Justin McMahon pushed forward for a rare point.
A lofted Cavanagh free clipped the tip of a post, but Hawk-Eye awarded the point as the Red Hands pushed two points clear.
Rock and Bernard Brogan, making his first start of the season, were on target to bring the sides level, but the opening quarter ended with rookie midfielder Padraig McNulty skipping past two defenders to fire the Red Hands back in front
Rock knocked over another couple of frees to being the home side level by the 32nd minute, and they went in level on 0-6 each at the break, but Tyrone were soon back in front with another Cavanagh free.
The Dubs pushed numbers forward, but were forced to shoot from distance as a compact Tyrone defences pressed incessantly.
The result was a rash of frustrating wides, and they were forced onto the back foot as the Red Hands attacked on the counter.
Some superb defending from Jonny Cooper denied Aidan McCrory and then Padraig McNulty, and having survived a sticky spell, Jim Gavin's side finally got their first score of the second half in the 45th minute through Emmet O Conghaile.
Tyrone were defending with discipline and breaking incisively, and once again opened up a two points lead with tidy points from Mattie Donnelly and Barry Tierney.
Rock and Paul Flynn were on target to bring the sides level, but once again, Mickey Harte's men rallied, and debutant Mark Bradley hit the target, along with McCurry, to make it 0-11 to 0-9 with ten minutes to play.
Justin McMahon and Cathal McCarron, until he went off injured, were supreme at the back for Tyrone, holding the Dubs scoreless for almost 15 minutes, until they struck for that sucker punch with a little over a minute to play.
Dublin: S Cluxton, E Culligan, R O'Carroll, J Cooper, N Devereux, P McMahon, J McCaffrey, D Bastick, E O Conghaile (0-1), T Brady, D Rock (1-6, 0-4f), C Kilkenny, K McManamon, P Andrews, B Brogan (0-1).
Subs: C O'Sullivan for Bastick (28), E O'Gara for Andrews (h-t), P Flynn (0-1) for Kilkenny (h-t), J McCarthy for McMahon (58), S Carthy for O Conghaile (63), C Costello for Brady (66).
Tyrone: N Morgan (0-1, f), A McCrory, R McNamee, C McCarron, R McNabb, J McMahon (0-1), P Harte, C Cavanagh, P McNulty (0-1), T McCann, M Donnelly (0-1), B Tierney (0-1), D McCurry (0-2, 2f), S Cavanagh (0-4, 4f), C McShane.
Subs: M Bradley (0-1) for McShane (h-t). PJ Lavery for McCaffrey (54), C McCann for Tierney (65).
Referee: D Gough (Meath).
Saturday 7 March
Football League Division One
Dublin 1-9 0-12 Tyrone
Cavan 0-14 2-13 Kildare
Meath 4-10 0-11 Westmeath
Armagh 1-10 1-10 Fermanagh
Tipperary 2-11 1-8 Clare
Sunday 8 March
Football League Division One
Cork 3-17 2-9 Kerry
Derry 1-13 2-12 Mayo
Donegal 1-4 0-9 Monaghan
Down 1-17 1-16 Galway
Laois 2-17 0-16 Roscommon
Sligo 2-26 2-9 Louth
Wexford 1-16 2-12 Limerick
Antrim 1-12 1-8 London
Offaly 2-13 1-16 Leitrim
Longford 1-12 1-10 Waterford
Carlow 0-9 0-9 Wicklow