The meeting of the teams placed top and bottom in Division 1 ultimately panned out as expected in Navan, Galway securing their third consecutive win and Meath slipping to their fifth loss in a row and, in the process, relegation.
It was far from straightforward for the visitors at Páirc Tailteann, though, as Padraic Joyce's men had to fight back from eight points down to secure the two-point win.
They slipped 1-6 to 0-1 down late in the first-half and looked in danger of a surprising loss after hammering Tyrone by 19 points last weekend.
At that stage, Meath supporters dreamed of possibly avoiding the drop with a glorious finish to the campaign before things turned sour for them.
Shane Walsh's seven-point haul was decisive for Galway and it was the captain that scored the last two points of the game to seal the win.
Paul Conroy had a huge impact from the bench too with four points and a key role in the 40th-minute goal scored by Ronan Steede which turned the game on its head.
The win leaves Galway clear at the head of Division 1 with two games to go, though Meath are officially relegated after failing to pick up a single point so far.
It won't get any easier for Andy McEntee's men either as they have intimidating trips to Dublin and Monaghan still to come.
They got a great start and took full advantage of the wind to lead by 1-6 to 0-1 after half an hour, the goal arriving in the 16th minute from the boot of Ethan Devine.
Galway finished the half strong with three converted frees from Walsh and an Eamonn Brannigan score to trail 1-6 to 0-5 at half-time and Steede's 40th-minute goal hauled them right back into it.
The sides were level three times up to 1-12 apiece late on but those two late Walsh points separated the teams and broke Meath hearts.
Galway scorers: Shane Walsh (0-7, 6 frees); Ronan Steede (1-1); Paul Conroy (0-4, 1 mark); Michael Daly and Eamonn Brannigan (0-1 each).
Meath scorers: Ethan Devine (1-0); James Conlon, Donal Lenihan (2 frees), Bryan McMahon (0-3 each): Bryan Menton (0-2); Oisin O'Brien (0-1).
Galway: Connor Gleeson; Sean Kelly, Sean Mulkerrin, Johnny Heaney; Liam Silke, John Daly, Gary O'Donnell; Ronan Steede, Tom Flynn; Eamonn Brannigan, Cein D'Arcy, Michael Daly; Martin Farragher, Shane Walsh, Adrian Varley.
Subs: Paul Conroy for Farragher (27); Gareth Bradshaw for O'Donnell (52); Robert Finnerty for Varley (55); Darragh Silke for Daly (61); Matthias Barrett for Steede (65).
Meath: Marcus Brennan; Robin Clarke, Conor McGill, David Toner; James McEntee, Ronan Ryan, Donal Keogan; Bryan Menton, Brian Conlon; Ethan Devine, Bryan McMahon, Cillian O'Sullivan; James Conlon, Oisin O'Brien, Thomas O'Reilly.
Subs: Shane Walsh for O'Brien (48); Donal Lenihan for Devine (55); Jack O'Connor for James Conlon and Padraic Harnan for Brian Conlon (55).
Referee: Niall Cullen (Fermanagh).