After their win over Wexford, Monaghan have been drawn against Carlow.
Neither Ulster side will have home advantage for the matches, which will take place on 15 of July.
In Round 4A, Donegal will face Galway, whose Connacht final loss to Roscommon meant there was no need for a draw.
Having already beaten Mayo in the provincial championship, Galway will avoid a repeat meeting with their neighbours.
It means Donegal will play Galway at Markievicz Park in Sligo on 22 July, while Mayo face beaten Munster finalists Cork on the same date at the Gaelic Grounds in Limerick.
Armagh are into the third qualifying round for the first time since 2014 following Saturday's five point win over Westmeath and their reward is a chance to gain revenge for their last-minute defeat by Tipperary, which cost them a place in Division Two of the National League.
The first championship meeting between the two counties will be held at Semple Stadium at 5pm, 15 July after Tipperary were the first team drawn from the pot this morning.
Following their victory in Wexford, Monaghan will face another Leinster opponent this weekend.
Carlow were comfortable 2-14 to 0-13 winners over Leitrim in Round 2B and they will once again have home advantage against Malachy O'Rourke's side.
The match will also be held on 15 July at Dr Cullen Park in Carlow with a 19:00 BST throw-in.
Armagh's qualifier win over Westmeath has earned them a rematch with Tipperary, who denied the Orchard footballers promotion from Division Three of the National League in April.