Paddy McBrearty settled a Pairc Tailteann epic with a stoppage-time winner to fire Donegal through to the final round of the All-Ireland football qualifiers.
The corner-forward nailed seven points on the evening though none was more important than his 73rd-minute winning score from long range.
It settled an enthralling game that had been level six times throughout the second-half and 11 times in total.
Michael Murphy weighed in with five points from his midfield station also while Ryan McHugh kicked 1-01 and the 61st-minute goal that looked to have finally put Donegal in the clear.
That strike put Donegal 1-12 to 0-12 ahead though Meath responded within seconds with a Cillian O'Sullivan goal themselves, setting up an exciting finale.
The sides were level twice more but McBrearty ultimately tipped the balance in Donegal's direction and they advance to play Cork or the Connacht final losers in Round 4A on July 22/23.
It's the end of the line for Meath this season and new boss Andy McEntee, who made a beeline for referee Derek O'Mahoney at full-time to remonstrate with the Tipperary official.
The hosts were noticeably more defensive than usual against Sligo in Round 2A last Saturday night and continued with the counter-attacking policy.
Donegal went with a similar game plan, dropping Murphy out of the full-forward line to partner Jason McGee at midfield.
Scores were hard to come by for both sides though there were some exciting moments when the two teams burst out of defence at pace.
The Leinster men improved on this aspect of their play from seven days earlier and punished Donegal on two significant occasions approaching half-time.
The visitors had forged 0-07 to 0-05 ahead and sought to extend their lead to five when Eoghan Ban Gallagher and Eoin McHugh raced through on goal in the 33rd minute.
But Gallagher's final pass for McHugh was intercepted by full-back Conor McGill and Meath broke free to score a point through Donal Keogan.
A similar scenario occurred in added time when Hugh McFadden had a Donegal goal ruled out for a square-ball offence and Meath restarted with gusto and won a free that Donal Lenihan converted.
That was Lenihan's third converted free of the half and left the sides tied at 0-07 apiece at half-time when, in truth, the Ulster side deserved to be at least three to the good.
Former All-Ireland winning captain Murphy scored two tremendous first-half points for Donegal, the second of which put Donegal ahead for the first-time in the 26th minute when he kicked long over Meath's massed defence.
McBrearty helped himself to three first-half scores and nipped in expertly for a 30th-minute point when Ruairi O Coileain was robbed in the Meath defence.
Michael Newman came in for Meath at half-time having scored two crucial late points against Sligo in what his first action of the year following ankle trouble.
He had a limited impact though Meath remained right in the game with points from Lenihan, Shane McEntee and James McEntee keeping the game perched in the balance.
McHugh finished off a great passing move with a palmed finish for Donegal's goal though O'Sullivan responded immediately for Meath when he bundled the ball over the line at the other end.
It looked all set for extra-time in a tit-for-tat encounter that went right to the wire but McBrearty showed all his experience and quality to nail the winner from all of 45 metres out.
Donegal: MA McGinley; C Ward, N McGee, K Gillespie; P McGrath, R McHugh (1-00), F McGlynn (0-01); J McGee (0-02), M Murphy (0-05, 0-02f); EB Gallagher, C Thompson, E McHugh; M O'Reilly, H McFadden, P McBrearty (0-07, 0-03f).
Subs: M McElhinney for Thompson 43, J Brennan for McFadden 48, K Lacey for O'Reilly 58, M McHugh for McGlynn 60.
Meath: P O'Rourke; D Keogan (0-01), C McGill, M Burke; P Harnan, B Power, S McEntee (0-01); B Menton (0-01), B Conlon; R O Coileain (0-01), J McEntee (0-01), G Reilly (0-01); S Tobin, C O'Sullivan (1-01), D Lenihan (0-07, 0-07f).
Subs: M Newman for S Tobin h/t, D Tobin for S McEntee 61, E Wallace for O'Sullivan 67 (black card).
Referee: D O'Mahoney (Tipperary).
