A run of eight unanswered points either side of half time sent Offaly on their way to a battling 1-12 to 1-9 victory over Westmeath in Tullamore.
Aaron Kellaghan's sixth-minute goal ultimately proved to be the decisive score for the hosts, who took Wicklow's scalp in the first round and will now carry a 100% record into their meeting with Dublin at this venue in seven days' time.
Offaly started and closed the first half strongly to remain on their opponents' heels at the break, 1-7 to 1-5.
After an even start, Kellaghan's major propelled the Faithful County into an early 1-1 to 0-2 lead, with Jordan McDonnell having slotted two frees for the losers, who also lost their opening round-robin assignment against Meath.
Martin McCabe's charges gradually settled into their stride and Jamie Coffey's 17th-minute goal helped them claim a four-point advantage with 18 minutes played, 1-6 to 1-2. The gap went out to five but the home side reeled off the last three scores of the opening period to stay in touch at the turnaround.
The comeback continued unabated upon the restart as Luke Egan and Mikey Cunningham struck to bring the Faithfuls level. At the three-quarters stage, Offaly led by 1-11 to 1-8, having hit nine of the last ten scores, McDonnell's free ending their sequence of eight without reply.
Jack Bryant, Sean Courtney (2) and Dan Lowry all posted precious points during that spell of Offaly dominance.
It went right down to the wire and Kellaghan pointed the winners into a three-point lead at the end of a flowing move on 57 minutes.
There was no way back for Westmeath as Offaly took both points on offer.
Offaly - S O'Toole; A Brazil, K Dolan, B Nolan; E Cullen, R Egan, H Balsiger; C Donoghue (0-1), M Tynan (0-1); L Egan (0-1), S Courtney (0-3f), M Cunningham (0-1); J Bryant (0-1), D Lowry (0-1), A Kellaghan (1-3, 0-2f). Subs: P Bennett for L Egan, L Pearson for B Nolan (BC), C Delaney for E Cullen, D Flynn for J Bryant.
Westmeath - S Fagan; C Drumm, C Daly, S Murtagh; J McDonnell (0-5, 4f), C Gibney, D Seery; A Cronin, J Coffey (1-1); E Gaffney, J Torpey (0-1), S Fleming; L Moran, K O'Sullivan (0-2), T Keena. Subs: E Martin for S Murtagh (BC), O Raleigh for T Keena, A Donoghue for L Moran, J Murtagh for J Coffey, E McCabe for J Torpey.
Referee - P McGuire.