A 49th-minute goal from substitute Adrian Maguire proved pivotal as Meath progressed to the next round with four points to spare over 14-man Dublin at O'Toole Park.
Playing with the wind in the opening period, the Royals raced into a five-point lead early on with scores from Ronan Lynch (2), Freddie Newman (2) and midfielder David Toner, but Dublin eventually settled to reduce the margin to the bare minimum with points from Adam Rogers (2), Killian O'Gara and Lee O Briain.
Sky Blues corner-back Conor Molloy was dismissed on a second yellow-card offence 24 minutes in, but a stoppage-time goal from O'Gara (brother of Dublin senior full-forward Eoghan) gave the hosts a 1-4 to 0-6 lead at the interval.
A point from Meath's powerful full-forward Peter O'Connell levelled matters on 33 minutes, but well-taken efforts from Rogers (2) and O Briain handed Dublin a two-point advantage heading into the final quarter.
Meath displayed tremendous resolve during this period, however.
Maguire's instinctive flick past Dublin netminder Kevin McGovern with 11 minutes remaining was added to by points from half-back Jamie Reilly, and the excellent double act of Lynch and Newman, as Meath pulled clear.
The home side's sole score during this juncture was a late pointed free from the accurate Rogers, with Meath taking control of the game.
SCORERS - Meath: R Lynch 0-5 (5f), F Newman 0-4 (3f), A Maguire 1-0, J Reilly, D Toner, P O'Connell 0-1 each. Dublin: A Rogers 0-5 (5f), K O'Gara 1-1 (0-1f), L O Briain 0-2.
MEATH – J Hannigan; M Flood, R McDonagh, A Leech; Jamie Reilly, C Carton, James Reilly; D Toner, G Kavanagh; R Moore, R Lynch, F Newman; S Maxwell, P O'Connell, N O'Neill. Subs: K Fenton for O'Neill (21), S Cocoe for Leech (h-t), A Maguire for Maxwell (40).
DUBLIN – K McGovern; C Molloy, I Bailey, K Tracey; S O Dulain, E Smith, B O Raghallaigh; D Casey, J Andrews; D Ryan, A Rogers, L O Briain; K O'Gara, S Rocks, K McAnerin. Subs: S Gleeson for O'Gara (h-t), C Grogan for Ryan (51), D Kiernan for O Briain (59).
REF – A Nolan (Wicklow).