Goals from Dean McGovern and Keith Beirne had the visitors on song in part two. An the end Galway were struggling and just about survived with their reward is a semi-final clash against Mayo in two weeks time.
Galway scorers- Robert Finnerty and Dessie Conneely 1-2 each, Cillian McDaid and Eoin Finnerty 1-0 each, Paul Mannion 0-2, Colin Brady, Peter Cooke, Michael Daly (sideline) 0-1 each
Leitrim scorers- Keith Beirne 1-4 (4fs), Dean McGovern 1-0, Mark Plunkett and Jack Heslin 0-2 each, Ryan O’Rourke, Robbie Cunningham, Conor Cullen, and Conor Gaffney 0-1 each.
Galway team - Ronan O’Beolain; Liam Kelly, Dylan McHugh, Ruairi Greene; Sean O’Ceallaigh, Sean Andy O’Ceallaigh, Kieran Molloy; Peter Cooke, Michael Daly; Colin Brady, Paul Mannion, Cillian McDaid; Robert Finnerty, Eoin Finnerty, Dessie Conneely. Subs: Conor Tarpey for Sean O’Ceallaigh (27), Colm Brennan for Peter Cooke (44), Cein Darcy for Eoin Finnerty (46), Antaine O’Laoi for Conor Tarpey (50), Michael Boyle for Robert Finnerty (57), Colm Mannion for Cillian McDaid (59).
Leitrim team- Jack Burke; James Mitchell, Keith Keegan, Liam Ryan; Donal Feely, Brian Leyden, Oisin Madden; Dean McGovern, Mark Plunkett; Jack Heslin, Conor Dolan, Conor Gaffney; Ryan O’Rourke, Keith Beirne, Robbie Cunningham. Subs: Conor Cullen for Ryan O’Rourke (7), Pearce Dolan for Donal Feely (19), Niall McGovern for Conor Dolan (27).
Galway enhanced their prospects of a return to Division One of the Allianz Football League for the first time since 2011 with three second-half goals in Pairc Esler.
Despite playing the final 25 minutes with 14 men following the straight red card dismissal of defender Declan Kyne, Galway dominated the second half and left Down needing a victory away to Cork next week to avoid relegation.
Two goals inside a minute from Eamonn Brannigan (a penalty) and Shane Walsh early in the second half put Galway in control, but Barry O'Hagan's 49th minute goal briefly gave Down hope of saving the game - and possibly their Division Two status.
Ryan Johnston almost grabbed a second goal for the home side, but Ruairi Lavelle made a superb save for Galway, who need a point against Kildare next week to make it back to the top flight after a seven-year absence..
The Tribesmen soaked up the pressure and counter-attacked really well, Gary Sice picking off a couple of fine scores on the break to take the sting out of Down's challenge.
The third Galway goal killed it. They were awarded a second penalty when Gary O'Donnell was upended in the square.
Down's sub goalkeeper Marc Reid, who came on when Michael Cunningham was black-carded in the first penalty incident, parried Brannigan's initial shot, but tapped home the rebound.
Down, conscious scoring difference could yet be crucial, battled to the end with Darragh O'Hanlon tapping over three late points to take his personal tally to 0-8.
The sides were level six times in a competitive and lively first half which ended 0-8 apiece.
Galway dominated the kickouts and took the initiative with early points from Barry McHugh and Shane Walsh in the opening three minutes and they had their noses in front for most of the opening half with Shane Walsh hitting three fine scores from play.
Down stayed with them, however, punishing Galway's over-elaboration in the final third.
Caolan Mooney nailed two great long-range points with the outside of the boot while O'Hanlon was on target from frees and play.
All the drama came in the third quarter with a flurry of goals and cards.
Gary Sice was fouled by 'keeper Cunningham, who was sent off, and Brannigan's shot beat his replacement Reid low to his left.
Good vision from Sean Armstrong set up Shane Walsh for a second Galway goal within 60 seconds. His reverse pass into the centre was met by the unmarked Walsh whose low shot went through a sea of legs to put Galway 2-8 to 0-8 ahead after 41 minutes.
Kyne's red soon after gave Down hope and O'Hagan's goal, after taking a high catch and out-foxing two defenders, had them back in it.
But three points was as close as they got and Lavelle's point-blank save from Johnston lifted Galway again and Brannigan's 68th minute goal clinched it for Kevin Walsh's side.
Final score: Galway 3-15 Down 1-13
Scorers for Galway: Eamonn Brannigan 2-1 (1-0 pen), Shane Walsh 1-3, Gary Sice 0-3 (1f), Barry McHugh 0-3 (2f, one '45'), Gary O'Donnell 0-2, Declan Kyne 0-1, Michael Farragher 0-1, Paul Conroy 0-1 (f).
Scorers for Down: Darragh O'Hanlon 0-8 (5fs), Barry O'Hagan 1-0, Caolan Mooney 0-2, Ryan Johnston 0-2, Shay Millar 0-1,
GALWAY: Ruairi Lavelle; David Wynne, Declan Kyne, Cathal Sweeney; Gary O'Donnell, Michael Farragher, Johnny Heaney; Paul Conroy, Fiontan O Curraoin; Thomas Flynn, Shane Walsh, Eamonn Brannigan; Gary Sice, Barry McHugh, Sean Armstrong.
Subs: Liam Silke for Wynne (39), Micheal Lundy for Armstrong (52), Danny Cummins for Walsh (68), Garreth Bradshaw for McHugh (70), Ronan Steede for Brannigan (70).
DOWN: Michael Cunningham; Ryan McAleenan, Brendan McArdle, Darren O'Hagan; Darragh O'Hanlon, Conaill McGovern, Caolan Mooney; Aidan Carr, Peter Turley; Kevin McKernan, Conor Maginn, Joe Murphy; Shay Millar, Barry O'Hagan, Ryan Johnston
Subs: Marc Reid for Cunningham (BC, 39), Cathal Magee for Carr (47), Jerome Johnston for Murphy (50), Paul Devlin for Millar (55), Jonathan Flynn for Mooney (BC, 70)
REFEREE: Paddy Neilan (Roscommon).