Meath registered half of their tally in the final quarter with David Bell posting two from play to go with another brace from frees.
The first of that quartet of scores restored Meath's lead. 0-5 to 0-4, in the 48th minute. Conor O'Donoghue's charges took that narrow advantage to the end.
Offaly's sole second-half score came from Cormac Delaney's 59th minute free that left it 0-7 to 0-5 to the home side.
Meath had to defend diligently in time remaining before Bell's fourth score - in the fourth minute of added time - settled matters.
While the scores were scarce and the quality low at times, supporters from both sides had much to shout about.
The win gives Meath - who lost to Dublin first time out - something to play for going into the final group games at home to Wexford next Wednesday. Offaly face Dublin on the same night, still in with a shout of reaching the business end.
On a damp (rain arrived in the latter stages), overcast and cool evening Meath finished with 14 wides to Offaly's one'
George Digan's side looked the stronger but faded in the latter stages though there never was more than a score between the teams.
Meath were quicker out of the blocks, posting three points within six minutes from the off. They failed to register a score over the remainder of the half.
Niall Finnerty posted the opening brace, his opening score the more notable. Simonstown's Josh Carolan provided the assist for Meath's third score, with Pearse Sheridan registering in the sixth minute.
The most alarming feature of Meath's first-half showing was their tally of nine wide.
Meath led by 0-3 to 0-2 at that stage. Only two more scores were posted over the remainder of the half. A Morgan Tynan free tied the scores for the first - and only time in the opening half - in the 20th minute.
Cormac Delaney put Offaly in front for the first time with his third pointed free of the half with the interval beckoning.
Offaly led by 0-4 to 0-3 after a mediocre opening half but had only one wide to Meath's nine.
Niall Finnerty's third point - within three minutes of the restart - levelled matters for the second and final time. The next score didn't register until the 47th minute but was the first of four final quarter scores for Conor O'Donoghue's charges.
Before the throw-in a minute's silence was observed in memory of the legendary GAA figure Eugene McGee who died at the weekend.
Meath – Dean Pluck; Jamie Fagan, Cillian Hegarty, Harry Corscadden; Sean O'Donoghue, Aaron Chinchilla, Ciaran Caulfield; Josh Carolan, Endi Ehichoya; Shane Smyth, Niall Finnerty (0-3), Eoghan Frayne; Pearse Sheridan (0-1), Barry Thompson, David Bell (0-4, 2fs). Subs - Evan Corrigan for Kiernan (42), Jamie Browne for Thompson (52), Jamie O'Shea for Finnerty (54), Adam McDonnell for Smyth (60 + 2).
Offaly - Cathal Ryan; David Carthy, Fionn Dempsey, David Dempsey; Sean Furlong, Adam Strong, Lee Pearson (C); Ryan Strong, Luke Egan; Aran Byrne, Cormac Delaney (0-4 fs), Adam Kelly; Harry Plunkett, Morgan Tynan (0-1 f), Darragh Flynn. Subs - Mark Bolton for Carthy (2), Jay Sheerin for Byrne (ht), James Nolan for Strong (48).
Referee - Gary Hurley (Westmeath)