Offaly boss John Maughan admitted that relief was his primary emotion as they survived a late Carlow comeback in Tullamore.
Leading by eight with 13 minutes to play, Offaly's margin was pegged back to two points before a late Cian Farrell score saw them over the line, 3-14 to 0-20, in wet conditions at O'Connor Park.
"There's no question about it [we're relieved]," Maughan told RTÉ Sport.
"We were [eight] points up there in the second half and you have to give great credit to Carlow, they kept plugging away.
"They were brave and it’s hard to believe they were struggling in Division 4 but they played some super football.
"We are a little bit disappointed with our execution, some of the fundamentals let us down. It’s an aspect of our game that we are concerned about, from our own kick-outs there was a problem as well.
"I’m just delighted that we won the game."
It was Offaly's first Leinster SFC win since 2016 and sets up a quarter-final meeting with Kildare in O'Moore Park next weekend.
"We’ve got to be better, we know that," added Maughan, who must wait to see how Shane Horan, Jordan Hayes and Aaron Leavy recover following first-half injuries.
"I like the fact that it’s a quick turnaround. I’m glad we are playing next weekend.
"The momentum is there. There is no pressure on us going to Portlaoise next Sunday playing Kildare and we’ll give it everything we have."