This was a real campaign decider for both these counties and it showed during a nervous conclusion when both sides squandered chances when they were deadlocked.
But at the end Paul Conroy had the composure to deliver a second win of the campaign for Galway and inflict a second one-point defeat on a Meath side whose League campaign is starting to flounder after two opening wins.
This game at Pearse Stadium may not have had championship intensity or importance, but there was a lot at stake and it was obvious from the reaction of the two managers afterwards.
"It leaves us with our backs against the wall regarding promotion," said Meath manager Seamus McEnaney. "The game against Derry next week is now a huge one for us.
"Up to now we have been playing very, very good football and we were playing high-intensity games and working extremely hard and making good decisions on the ball. Last week's one-point defeat was bitter to swallow but at the same time it was a big performance.
"But this was a disappointing performance, especially the first half. We seemed to be only in second gear. The second half we had enough possession to push on and win the game, maybe we didn't take the right choices at the right time."
Galway manager Alan Mulholland, whose side's opening-day win over Derry was followed by a draw against Louth and a shock loss to Westmeath, said that the victory can alter their campaign hugely.
"It's a big, big win for us," said Mulholland. "We had two poor performances against Louth and Westmeath and we were afraid that we might get sucked into a bit of a spiral.
"What was great was that we got the response from everybody. We played with a lot of intensity, a lot of effort, which the team were criticised for last week, and probably rightly so.
"We showed great composure in that we worked the ball through -- in the last couple of weeks we have been going for silly scores in unreachable areas of the pitch -- and showed the composure that was missing in the last two games.
Conroy, who captained the 2007 All- Ireland minor-winning team which Mulholland managed, was the hero yesterday as he fired home the goal which separated the sides for long periods and then hit a dramatic winner two minutes into stoppage-time after a controversial Joe Sheridan goal levelled the game.
The Seneschalstown man fisted to the net nine minutes from time when he latched on to a high delivery from Cian Ward which seemed to be sailing wide.
Referee Gary McCormack consulted with his umpires before awarding the goal despite claims from Galway that Sheridan was in the square and that the ball was wide.
That levelled the game and ensured a tight finish where both sides squandered good chances before Conroy had the composure and confidence to have a go from 35 metres and land the winner.
Conroy also supplied the main score of the opening half when he showed Meath full-back Kevin Reilly a clean pair of heels to solo in from the left and drill the ball low to the net from a tight angle.
That helped Galway open up a 1-7 to 0-6 interval lead after the sides had been level four times before Conroy's goal.
Nicky Joyce was in superb form for Galway and landed 0-7, four of them from play, and that helped the Tribesmen open up a five-point lead 10 minutes after the restart.
But Meath struck back and hit three points in a row, with Cian Ward getting a second '45' and Mark Ward hitting another superb effort from distance, before Sheridan found the net.
Galway drafted in Padraic Joyce for his 15th campaign and they secured an invaluable win when Conroy hit the winner in the dying moments.
Man of the Match: Nicky Joyce (Galway)
Scorers -- Galway: N Joyce 0-7 (3f), P Conroy 1-1, D Cummins, N Coleman, G Bradshaw, G Sice 0-1 each. Meath: S Bray 0-4 (4f), C Ward 0-3 (2'45), J Sheridan 1-0, M Ward 0-2, S McAnarney, T Walsh 0-1 each.
Galway -- A Faherty 7; K McGrath 7, F Hanley 7, K Kelly 7; G Bradshaw 8, D Blake 6, G O'Donnell 6; G Higgins 6, J Bergin 7; G Sice 7, D Burke 8, N Coleman 6; N Joyce 9, P Conroy 8, D Cummins 7. Subs: T Flynn 6 for Coleman (59), J Duane for Blake (64), P Joyce for Cummins (67), M Hehir for Burke (70).
Meath -- D Gallagher 6; M Burke 6, K Reilly 6, B Menton 6; C Lenihan 6, S McAnarney 6, D Toibin 7; T Walsh 7, M Ward 8; C O'Connor 6, G Reilly 6, S Kenny 7; P Gilsenan 6, C Ward 7, S Bray 7. Subs: C McGuinness 6 for Lenihan (27), J Sheridan 7 for O'Connor (30), G O'Brien 7 for Toibin (h-t), J Queeney 6 for Reilly (50), B Meade for Kenny (73).
Ref -- G McCormack (Dublin).