Down booked a place in the Christy Ring Cup decider against Kerry thanks to a strong second-half display in the semi-final replay at Ballycran.
Mark O'Sullivan netted for Meath but the hosts led 0-12 to 1-6 at the break before taking control .
Meath finished with 13 men after defenders Cormac Reilly and Sean Heavey were sent-off while Down's Paudie Flynn was shown a straight red late on.
Paul Braniff fired over eight points for the Mourne side.
Meath opened this game brightly with an O'Sullivan point, but Down hit back with scores from Braniff and Danny Toner.
Meath regained the lead thanks to the only goal of the game - an O'Sullivan effort on eight minutes, after the Kiltale clubman cut inside and smashed the ball past Down keeper Stephen Keith.
Little separated the sides for the remainder of the opening quarter, with a superb score from midfielder Ciaran Coulter levelling the tie at 0-6 to 1-3 after 16 minutes.
Down rued six wides at this stage, with Meath moving ahead once again.
Stephen Clynch added back-to-back scores to ensure a 1-5 to 0-7 lead for the visitors with 11 minutes remaining in the opening half.
However, Down proved strongest in the closing stages of the half with four unanswered points between the 25th and 34th minutes.
Braniff, Fintan Conway and James Coyle on target to open up a three-point lead, with Clynch and Braniff trading scores before the break to ensure a three-point interval lead for Down.
Scores were at a premium in the third quarter, as defences got on top.
Eoin Marsh broke the scoring deadlock for the visitors, but his point was quickly cancelled out by Braniff's sixth score.
The Down captain's effort kick-started a magnificent spell by the hosts, who went on to add six unanswered points.
Paul Sheehan scored three spectacular points from play, with scores also from Braniff, Johnston and two excellent efforts from Kevin McGarry to ensure a 0-19 to 1-7 lead entering the final quarter.
Reilly and Heavey were sent to the line for two yellow cards in the closing minutes before Flynn was dismissed.
Scorers for Down: P Braniff 0-8 (6f, 1 '65); K McGarry, P Sheehan 0-3 each; F Conway, D Hughes, D Toner 0-2 each; C Coulter, G Johnston, J Coyle 0-1 each.
Scorers for Meath: S Clynch (3f) 0-5; M O'Sullivan 1-1; S Morris, J Kelly, E Marsh, J Meyler 0-1 each.
Down: S Keith; M Conlan, P Flynn, P Hughes; F Conway, C Woods, M Ennis; D Hughes, C Coulter; P Sheehan, P Braniff, S Nicholson; D Toner, G Johnston, J Coyle. Subs: K McGarry for Nicholson (40); M Turley for Coyle (60); J McCusker for Toner (65); E Clarke for Sheehan (66); B Nicholson for Coulter (69).
Meath: S McGann; D Donoghue, D Healy, C Reilly; J Toher, M Burke, S Heavey; S Donoghue, S Morris; E Marsh, S Clynch, J Kelly; P Durnin, M O'Sullivan, P Conneely. Subs: B Slevin for Conneely (55); J Meyler for S Donoghue (58); A Gannon for Durnin (66).
Referee: Tony Carroll (Offaly)
In the Leinster Senior Football Championship, Kildare and Dublin will meet in the semi-finals.
Kildare beat Offaly 0-19 to 1-12 with Johnny Doyle scoring six points for Kieran McGeeney's Lilywhites.
In the second quarter-final of the Croke Park double-header, Dublin overpowered Westmeath by a margin of 1-22 to 0-09, Bernard Brogan getting seven of the Dubs' tally.