The visitors were the better team in the opening half and after trading early scores, they surged ahead with six points without reply in a seven-minute spell.
That ultimately proved to be the winning of an otherwise tight contest, although Louth did close the gap to a more manageable four points at half-time, 0-8 to 0-4.
A fortuitous goal from leading marksman Jim McEneaney brought the hosts right back into the contest, but Joe Maher kept Offaly ticking over, and they held on for a narrow victory.
B Allen 0-4, S Doyle (3f), J Maher (2f) 0-3 each, C McNamee, M Brazil, C Donoghue 0-1 each.
J McEneaney 1-5 (4f), C Judge 0-2 (1f), G McSorley, R Holcroft 0-1 each.
K Garry; D Brady, E Rigney, S Pender; N Darby, J Moloney, J O’Connor; L Fox, C McNamee; M Brazil, S Nally, C Donoghue; S Doyle, J Maher, B Allen.
J Gettings for Fox (43), A Sullivan for Brazil (43), C Byrne for Darby (53), N Dunne for Doyle (55), S Moriarty for Nally (60), B Darby for Pender (68).
N Gallagher; P Rath, D Quigley, K Toner; E Lafferty, D McMahon, K McLaughlin; T Durnin, D O’Connell; A Reid, J McEneaney, D Byrne; C Judge, G McSorley, J Stewart.
C McKeever for O’Connell (30), R Holcroft for McSorley (50), C Lenehan for Toner (50), J Califf for Durnin (56), S Hand for Judge (69).
B Cawley (Kildare)