The scores were tit for tat after a lightning start which yielded green flags for both sides inside six minutes.
The teams were locked midway through the second half, but Mallow pushed on after Donal Relihan netted their third goal — a spectacular fisted effort eight minutes from time.
Nemo Rangers, who must now play Castletownbere in the second round, will take heart from this spirited performance.
They opened with aplomb when corner-forward Conor Allen had the ball in the net after just 40 seconds. Mallow were not to be outdone though, and with James Loughrey blasted a great goal.
Nemo went two points clear but the Mallow response was impressive, hitting three points before Ryan Harkin goaled. When Loughrey punched over a point on 20 minutes, the Avondhu side were 2-4 to 1-2.
Nemo found flags hard to come by thanks to solid defending by Mallow’s Matthew Taylor, but got through for a crucial Alan O’Donovan goal.
However, Mallow closed out the half with a pair of points from the tireless Killian O’Connor and a Cian O’Riordan free.
Playing with the wind, three Conor Horgan frees and an excellent punt from Brian Twomey put Nemo on level terms by the 40th minute.
Loughrey pointed to edge his side ahead, but another brace from Conor Horgan saw Nemo hit the front.
O’Connor then worked the ball up to Relihan who netted to put Mallow back in control.
Shane Martin brought it back to the minimum and extra-time loomed.
However, Mallow rallied once more.
O’Connor drew a splendid save from Michael Dorgan and when the ball was worked back in, Cian O’Riordan landed a point before O’Connor put Mallow three points to the good with as many minutes remaining.
Conor Horgan kicked his sixth point to close the gap to two but Mallow had done enough.
Scorers for Mallow: J Loughrey 1-3, D Relihan 1-1, R Harkin 1-0, C O’Riordan (1f) and K O’Connor 0-2 each, P Herlihy 0-1.
Scorers for Nemo Rangers: C Horgan 0-6 (5fs), A O’Donovan 1-1, C Allen 1-0, D Kearney, B Twomey and S Martin 0-1 each.
MALLOW: C Sheehan; B Sheahan, K McCarthy, M Taylor; L Dockery, C Kerrisk, P O’Riordan; H Dockery, J Loughrey; C O’Riordan, E Stanton, P Herlihy; R Harkin, D Relihan, K O’Connor.
NEMO RANGERS: M Dorgan; J Garvin, P Brophy, S McKeown; S Martin, E Weldon, C O’Shea; D Mehigan, S Twomey; J Horgan, B Twomey, A O’Donovan; C Allen, D Kearney, C Horgan.
Referee: M Moynihan (Grenagh).