Mickey Harte's side came out just on top after an enthralling second half when the lead changed hands many times.
Tyrone hit four unanswered points in a devastating spell near the end through Plunkett Kane (2), Patrick McNeice and Peter Harte to leave Dublin with too much to do in the final minutes.
Tyrone lead 0-10 to 1-6 after a first half of outstanding quality, with Philly Ryan getting the Dubs' goal.
But two points from the outstanding Seán Cavanagh, and a fine score from Stephen O'Neill brought Tyrone back and they finished the half the stronger of the two sides, leading by one at the break following two more scores from the same two players.
O'Neill, Cavanagh and McManamon added further points early in the second half but Dublin then went ahead after four points from Diarmuid Connolly, Paul Flynn, McManamon and Paddy Andrews to trail by a single score.
But then came the Tyrone purple patch of four points without reply.
Dublin kicked two points before the end, from Connolly and substitute Michael Darragh Macauley but it wasn't enough to deny the Red Hands their fourth win in five starts.
Meanwhile Down remain in relegation trouble after losing 0-11 to 1-5 to Kerry in Tralee as the Kingdom secured their first win of the campaign.
Kerry led by 0-8 to 0-1 at the break, with Connaire Harrison scoring the visitors' only point over the 35 minutes.
Down made the perfect start to the second half when the in-form Donal O'Hare found the net three minutes after the restart.
The visitors quickly added another score, but Kerry quickly built up a five point lead despite missing a penalty.
Brian Maguire nudged Kerry further ahead with nine minutes left and the Kingdom held on in the closing stages to register their first league points of the season.
In Division Two Armagh fell to a 1-12 to 1-10 home defeat against Louth at the Athletic Grounds where the visitors kicked over the final three points of the game in injury time.
The win gave Louth manager Aidan O'Rourke, the former Armagh player, added satisfaction as his charges take another crucial step on the road to survival in the second tier.
The sides were level at 1-5 apiece at the break, with Tony Kernan finding the net for the home side.
The Orchard County took the lead when Stefan Forker hit the target after the restart and things were looking even better when Louth lost Gerard Hoey to a red card but Armagh's James Donnelly followed him to the dressing rooms shortly afterwards.
Brian White sent a penalty wide seven minutes from the end but Armagh regained the lead with six minutes to go before Louth sealed the win with that late rally.
Also in Division Two Derry were emphatic 0-14 to 0-5 winners over Longford at Celtic Park.
Following last Saturday's draw with Armagh, Derry are back on course for promotion while Longford have now lost five out of five.
The Oak Leaf County were 0-5 to 0-2 ahead at the break after a hat trick of scores from James Kielt.
Eoin Bradley also sent over three points after the break when Raymond Wilkinson hit two as derry ran out easy winners.
Meanwhile in Division Three, Fermanagh were 1-12 to 1-10 winners over Cavan at Breffni Park despite having three men sent off.
Daniel Kille's late penalty was the crucial score for Fermanagh, who had Marty O'Brien and
Declan McCusker shown red cards late on after the earlier dismissal of Barry Owens.
Antrim's promotion hopes were dealt a major blow at Casement Park, where they were beaten 0-11 to 1-6 by Roscommon.
Frank Dawson's side trailed 0-7 to 0-1 at the break, with Anto Healy's late goal merely adding some gloss to the scoreline.
Allianz Football League Results:
- Tyrone 0-18 Dublin 1-14
- Kerry 0-11 Down 1-5
- Louth 1-12 Armagh 1-10
- Derry 0-14 Longford 0-5
- Cavan 1-10 Fermanagh 1-12
- Antrim 1-6 Roscommon 0-11