After withstanding initial Meath resistance in the opening half, odds-on favourites Kerry eventually came good in a battling encounter at Austin Stack Park.
Kerry struggled for long periods after leading 0-13 to 0-10 at half-time.
Meath enjoyed their best spell in the third quarter as points from Nicky Horan, Michael Cole and Peter Durnin saw them twice lead briefly by a point.
When Horan pointed the Royals in front from a 55th-minute free (0-17 to 0-16), an upset looked on the cards. However Kerry upped the tempo and a 65th-minute goal from John Egan put them six points clear.
From there they held on to reach the semi-finals, despite an injury-time goal from Michael Cole.
Man of the Match: T Murnane (Kerry)
Scorers -- Kerry: D O'Connell 0-7 (6f), J Egan 1-1, S Nolan 0-4, D Dineen, M Boyle, J Flaherty 0-2 each, G O'Brien, A Boyle 0-1 each Meath: N Horan 0-7 (6f, 1 '65'), M Cole 1-1, J Keena, P Durnin 0-3 each, G O'Neill, N Kirby, S O'Donoghue 0-1 each
Kerry -- B Rochford 7; J Godley 7, T Murnane 9, E Fitzgerald 8; J Casey 8, L Boyle 8, P O'Grady 7; D Dineen 8, J Egan 8; M Boyle 7, G O'Brien 6, S Nolan 7; D O'Connell 6, A Boyle 6, J Flaherty 7 Subs: A Royle 7 for A Boyle (h-t), P O'Keeffe 6 for S Nolan (52).
Meath -- S McCann 7; D O'Donoghue 6 , E Fitzgerald 7, W Mahady 7; M Horan7, P Fagan 8, D Kirby7; P Keogh 6, G O'Neill 7; J Keena 8, N Kirby 6, M Cole 8; D Raleigh 5 , N Horan 7, P Durnin 7. Subs: P Geoghan 7 for D Raleigh (28), S O'Donoghue 8 for P Keogh (h-t), E Keogh 6 for G O'Neill (65).
Ref -- C Lyons (Cork).