Mayo led 0-10 to 0-9 at the interval, with Kenny Feeney once again the main man in the scoring stakes.
The Royals opened up a four point gap midway through the second half, but a succession of points from Feeney saw Mayo draw level five minutes from time. Late chances fell to both teams before Clynch landed the late free to edge them into the semi-finals.
Scorers for Meath: S Clynch (0-8, 7f), C King, W McGrath (0-2 each), S Whitty, S Heavey, J Meyler, D Doran (0-1 each).
Scorers for Mayo: K Feeney (0-10, 8fs), F Boland (0-2), D Gallagher, K Higgins, S Regan (0-1 each).
MEATH: C Bird; C Reilly, N McLoughlin, D Donoghue; P Farrell, J Toher, S Whitty; S Heavey, D Reilly; C King, S Clynch, J Meyler; W McGrath, D Healy, S Morris.
Subs: D Doran for D Reilly, S Doyle for D Healy, J Regan for J Meyler.
MAYO: D O’Brien; A Brennan, A Connolly, D Kenny; D Gallagher, C Charlton, P Barrett; G Nolan, B Higgins; K Higgins, D McDonnell, F Boland; K Feeney, S Regan, B Hunt.
Subs: J Kennedy for B Higgins (inj), E Collins for D McDonnell, S Hoban for B Hunt, D McTigue for F Boland.
Referee: John O’Brien (Tipperary).