Captain Graham Reilly led by example with seven points at Parnell Park as Meath braved a dramatic Leinster quarter-final encounter to heap further misery on their neighbours and rivals.
Louth led by two points with 25 minutes to go after being gifted a bizarre goal following an error by Meath goalkeeper Paddy O'Rourke.
But stuttering Meath finally got their act together and finished with a barrage of scores to win with plenty to spare, setting up a semi-final clash with Kildare on 17 June.
It was a first win in the Championship for new Meath manager Andy McEntee, though he will feel there's room for improvement despite the huge point haul.
Meath had a string of goal chances but blew the lot while Louth, famously beaten by Meath in the 2010 Leinster final, netted three times in an enthralling end-to-end encounter.
There was intrigue before the throw-in as Meath were forced into an unexpected late change.
James McEntee, nephew of boss Andy, injured his hip in the warm-up and James Toher, last year's Christy Ring Cup-winning captain, started instead.
It proved a blessing in disguise as Toher had a strong first half, kicking four points and nailing two excellent scores from open play as he went on to hit 0-05.
Skipper Reilly started strong too and blasted a brace of points in the opening 10 minutes.
Meath moved 0-05 to 0-02 clear and had a third-minute goal chance when Donal Keogan's shot was well saved.
They cut through Louth's defence with ease and had three more gilt-edged goal chances in the first half but Cillian O'Sullivan, Bryan McMahon and Donal Lenihan wasted them all.
Meath still led by 0-08 to 0-04 after 23 minutes but 1-01 from Louth's Jim McEneaney tied the game up.
McEneaney, one of three late additions to the Louth line-up, hit a great point before nailing a 29th-minute penalty after a deliberate body check by Meath full-back Conor McGill which warranted a black card.
Donal Lenihan responded with two points to leave Meath marginally ahead at the break, 0-10 to 1-06, but the drama was only just beginning.
Louth were excellent in the third quarter and scored a bizarre 39th-minute goal to take a 2-06 to 0-11 lead.
Ronan Holcroft intercepted a Meath kick-out and backpedalling Meath keeper O'Rourke appeared to lose his bearings, leaving the goal wide open for Holcroft's miscued point attempt to drop straight in.
It was a giant stroke of fortune for Louth, who moved two clear and also scored a third goal in the 58th minute through Conal McKeever.
But even at that Louth couldn't take full advantage as Meath were much improved in the last 20 minutes or so and opened up with some excellent play to take control of the game.
Reilly was terrific when Meath needed him most and picked off a series of scores as the Royals outscored Louth by 0-14 to 1-00 from the 48th minute to win at their ease.
Substitutes Thomas Reilly and Ruairi O Coileain both contributed a brace of points after coming on while Sean Tobin was on the mark with a fine score too.
Meath: P O'Rourke; D Tobin, C McGill, M Burke; P Harnan, D Keogan, S McEntee; B Menton, R Jones (0-01); J Toher (0-05, 0-03f), C O'Sullivan (0-01), E Wallace; G Reilly (0-07), B McMahon (0-02), D Lenihan (0-06, 0-03f, 0-01 45).
Subs: D McQuillan for McGill (30, black card), R O Coileain (0-02) for S McEntee (52), C O'Brien for Menton (55), T Reilly (0-02) for McMahon (59), S Tobin (0-01) for O'Sullivan (69).
Louth: C Lynch; P Reilly, K Carr, P Rath; J Bingham, L Dullaghan, A Williams (0-01); T Durnin, J Stewart; D Byrne, J McEneaney (1-02, 1-00 pen), B Duffy (0-01); P Smith, E O'Connor, R Burns (0-02, 0-02f).
Subs: K Murphy for Rath (20), A McDonnell for Byrne (26), R Holcroft (1-00) for Carr (29, black card), C McKeever (1-00) for Dullaghan (h/t), J Califf (0-03, 0-01f) for Durnin (41), R Moore for Smith (70).
Referee: B Cassidy (Derry).
