Donegal were far from impressive against a very poor Roscommon team in Ballybofey on Saturday evening.
They had the luxury of a great start but allowed the Connacht men back within two points at the break.
Roscommon were reduced to 14 when Cathal Cregg received a straight red and five minutes into the second half they lost Johnny Dunning, who was side-lined along with Neil McGee after the pair became embroiled in a scuffle.
Donegal were ahead by two goals inside eight minutes. Michael Murphy was on the end of the first while he set up new boy Stephen Griffin for the second.
Roscommon got back in the game on 17 minutes when Karol Mannion drove home a penalty. Further points from Ger Heneghan saw Donegal ahead by just two at the break, 2-3 to 1-4.
Early in the second half they were reduced to 13 and their challenge withered and Donegal’s Michael Murphy found the corner from a penalty to end the game as a contest.
Donegal manager, Brian McIver, was relieved with the win.
He said: "It was a job well done as far as I'm concerned. The important thing for us is that we are into the next round.
"With no disrespect to Roscommon we will face tougher tests from here on in. But at least we came through and I thought we showed character and held our composure when they came back at us in the closing minutes of the first half."
DONEGAL: P Durcan; E McGee, N McGee, K Lacey; F McGlynn, B Monaghan, K Cassidy; N Gallagher, R Kavanagh (0-1); K Rafferty(0-1), D Walsh (0-1), S Griffin (1-0); C McFadden (0-5m3f,45), R Bradley, M Murphy (2-2,pen,2f).. Subs., S McGowan for Monaghan; B Devenney for Bradley; C Toye (0-1) for Walsh; M McGowan for McGlynn; C Dunne for Walsh.
ROSCOMMON: G Claffey; P O'Connor, D Casey, S McDermott; D Keenan, G Cox (0-1), S Daly; M Finneran, K Mannion (1-1,pen); M O'Carroll, D O'Gara, J Rogers; C Gregg, S Kilbride (0-1), G Heneghan (0-5,5f). Subs., J Dunning for Kilbride; S O'Neill (0-1) for Finneran; A McDermott for Keenan; C Devaney for Rogers.
Referee: J Bannon (Longford)