Cavan and Armagh will have to meet again as they played out a 1-14 to 0-17 draw after extra-time in Clones.
The replay will take place at the same venue next Sunday as a double-header with the first-round Qualifier between Monaghan and Fermanagh.
Aidan Nugent fired Kieran McGeeney’s side ahead for the first time in the 13th minute, Burns and Rian O’Neill having cancelled out frees from Gearoid McKiernan and Niall Murray.
McKiernan’s second pointed free put Cavan ahead, Armagh responded immediately as Burns curled over an excellent point.
A wonderful Conor Moynagh point off the outside of the right boot-tied the scores up for the fourth time.
Murray’s second converted free edged Mickey Graham’s charges back in front but O’Neill whipped over a tricky free on 20 minutes to make it five apiece.
Gerard Smith sent over a terrific score, also off the outside of the boot, but substitute Joe McElroy replied straight away.
Cavan suffered a big blow eight minutes from the break when full-back Padraig Faulkner was black-carded, albeit with O’Neill missing the resultant free.
Cavan wouldn’t hit the front again. After a long spell without a score at either end, O’Neill’s 35th minute free had Armagh in front but Christopher Conroy having replaced the injured Smith, equalised deep into injury time following good work by McKiernan.
The sides were level twice more upon the restart, Dara McVeety and Murray took it in turn to cancel out points from Paul Hughes and Jemar Hall.
The game swung into Armagh’s favour on 40 minutes Jarlath Og Burns got forward from midfield and planted the ball in the net after being picked out by Jamie Clarke.
O’Neill’s free into the wind increased the gap but Murray pointed a brace of frees either side of an Aidan Forker reply as three points separated the teams.
Cavan edged closer Stephen Murray’s strike on 54 minutes, the No.21 having only entered the fray two minutes earlier.
Nugent banged over a reply off his left peg but, with 11 minutes remaining, Stephen Murray turned down a goal chance and once more reduced the arrears to a couple of points, setting up a dramatic finale.
Clarke spurned a chance to increase the Orchard advantage but Cavan’s hopes took a hit with nine minutes remaining, Ciaran Brady received a straight red card for catching Mark Shields with a swinging arm.
However with six minutes of normal time left, McVeety showed tremendous character to close the gap to one point, substitute Cian Mackey equalising with a brilliant strike from 45 metres 60 seconds later.
Clarke tapped over a close-range free to restore Armagh’s lead but the 14 men hit back immediately through Mackey again at the end of a terrific team move.
Armagh had Brendan Donaghy sent off for a second yellow-card offence four minutes into stoppage time and it finished all level after normal time.
As the tension rose, the next score didn’t arrive until the third minute of the second period of extra time, Stefan Campbell broke free to fist the Orchard County back in front.
Despite having had Conor Moynagh black-carded, Cavan equalised through Mackey once again, another magnificent score three minutes from the end of extra time.
They will do it all again next Sunday in Clones with an Ulster-final place up for grabs.