James Horan admitted Mayo are "not where we want to be" after their defeat away to Down on Saturday night left them struggling in the league.
Down led 0-9 to 0-4 after 40 minutes in Newry, but the visitors fought their way back to level the scores at 0-11 apiece through a '45' from goalkeeper Ken O'Halloran with five minutes to go.
The game was in the melting pot and it was Down – who lost out to an injury-time goal against Cork last week – who finished the stronger.
For the second week running, Donal O'Hare kicked nine points for Down and his free in the last minute of normal time nudged the Mournemen ahead.
He added a point a minute later to ensure Down collected their first league points. The free was moved forward after a push in the back from Mayo sub Kevin Keane, which disappointed his manager.
"With two minutes to go we had come back to level and we had all the momentum, we were playing the football," Horan said.
"Down got an attack and there was a little shove on the back and there was a free in. There were three of those in the last couple of minutes so that is disappointing, but we'll learn from it. We have a couple of home games to come, so we will look forward to those."
Mayo full-back Ger Cafferkey suffered a bad nose injury after a poor frontal challenge from Conor Laverty towards the end of the first half.
The same player also scored a contentious point in the 24th minute, which looked wide, but after some hesitation the umpire on the near side raised the white flag.
Mayo hit five wides in the opening 20 minutes. Although they led 0-4 to 0-2 after a point from the impressive Cathal Carolan in the 16th minute, they failed to score again for the remainder of the first half.
Mayo twice carved a porous Down defence open in the first quarter, but on both occasions Jason Doherty and Michael Conroy opted to fist the ball over the bar when the goal chance was on.
Down then reeled off seven points in a row either side of half-time. They scored four without reply in the 15 minutes before half-time to lead 0-6 to 0-4 at the break.
Down extended their lead with three quick-fire points in the first five minutes after the restart.
Their 'keeper Brendan McVeigh, back in favour and making his first league start, made a superb reflex save from Barry Moran in the 38th minute. Lee Keegan popped a pass up from the ground and he steamed on to it, looking certain to bust the net, but McVeigh saved from point-blank range. Mayo were held scoreless for 26 minutes either side of the interval, but Carolan ended the drought in the 41st minute.
Barry Moran then found his range before Carolan scored a superb point with his left foot despite being off balance.
Moran and Aidan O'Shea completely took over in midfield and Mayo began to dominate. From 0-9 to 0-4 behind after 40 minutes, they levelled it up at nine points apiece 17 minutes later with a Jason Doherty free.
Having blown a lead last week, Down looked nervous as Mayo sensed blood, but O'Hare steered them home. A relieved James McCartan said: "Obviously we felt we should have won last week. It would have been easy to feel sorry for ourselves, but we told the lads if they kept putting in performances like that, the results would look after themselves.
"We didn't play as fluidly as we did last week, but it was very nice to win a tight match."
Man of the match: D O'Hare (Down)
Scorers – Down: D O'Hare 0-9 (7f), R Mallon, C Laverty, M Poland, P McComiskey 0-1 each. Mayo: J Doherty 0-4 (3f), C Carolan 0-3, M Conroy, B Moran, K McLoughlin (f), K O'Malley ('45') 0-1 each
DOWN – B McVeigh 8; D McCartan 7, B McArdle 6, R Boyle 6; R Mallon 6, A Carr 8, K Quinn 6; K King 5, K McKernan 6; C Laverty 6, M Poland 6, P McComiskey 6; D O'Hare 9, C Harrison 6, B Coulter 8. Subs: D O'Hagan 5 for Mallon (45), J Johnston 5 for Quinn (54), C Gough 5 for King (62), E McCartan 5 for Harrison (62).
MAYO – K O'Malley 7; C Barrett 6, G Cafferkey 7, K Higgins 7; L Keegan 7, D Vaughan 7, C Boyle 7; B Moran 9, A O'Shea 8; K McLoughlin 6, R Feeney 5, C Carolan 8; E Varley 6, J Doherty 7, M Conroy 6. Subs: K Keane 5 for Cafferkey (35), T Cunniffe 5 for Feeney (43), A Freeman 5 for Vaughan (54), E Regan 5 for Varley (63)
REF – D Coldrick (Meath)