Fulham had to come from behind at home to earn a replay in an entertaining 1-1 draw with Championship club Sunderland.
The visitors took a shock lead in just the sixth minute when Jack Clarke capitalised on Issa Diop's mistake which inspired the Black Cats to a hugely impressive performance at Craven Cottage.
Tony Mowbray's side racked up 17 shots but couldn't find another way past goalkeeper Marek Rodak while Fulham also laid siege with 21 efforts - although were denied time and time again by the outstanding Anthony Patterson.
- How the teams lined up | Match stats
- FA Cup fixtures | Results
- Download the Sky Sports app | Get Sky Sports
Cairney, though, did equalise for Fulham with a wonderful solo goal that turned the Sunderland defence inside and out before lashing into the bottom corner.
The visitors thought they had won the tie when 15-year-old substitute Christopher Rigg scored in injury time, but the goal was ruled out for offside.
How the entertainment unfolded
Fulham started the game on the front foot as they looked to return to winning ways following successive 1-0 defeats to Tottenham and Newcastle. Manager Marco Silva made seven changes to his starting XI, but it was Sunderland who took an early lead.
Clarke capitalised on a mistake from Issa Diop, taking the ball off the Fulham defender before running in on goal and slotting past Marek Rodak.
Fulham almost equalised in the 14th minute but Tosin Adarabioyo's header from an Andreas Pereira cross bounced into the ground and over goalkeeper Anthony Patterson and the crossbar.
Ross Stewart was then carried off the field on a stretcher with what appeared to be an ankle problem, forcing Sunderland into their first substitution of the game and denying him the chance of adding to his tally of 10 goals in 13 games this season.
Fulham had another good chance on the stroke of half-time from a quick run down the left wing from Manor Solomon, who pulled the ball back to Pereira but his shot was expertly blocked on the line by Daniel Ballard.
Harry Wilson also had an opportunity for the hosts when he was picked out by Cairney on the right-hand side, cutting inside to create space to shoot and forcing Patterson into a good save.
Patterson denied Wilson again minutes later when he came off his line to close the angle for the Fulham winger.
Sunderland almost doubled their lead when they caught Fulham on the break, with Patrick Roberts winning the ball, running through most of the home side's half before taking a shot, which was just wide of Rodak's goal.
However, Fulham eventually levelled from a clever piece of individual skill from Cairney just after the hour mark.
The midfielder was picked out by Layvin Kurzawa in the box but still had a lot of work to do, weaving his way past two defenders to create space before firing past Patterson.
Silva brought on Aleksandar Mitrovic and Willian in an attempt to force a finish, and Fulham's leading goalscorer did send a header straight at Patterson from a Kenny Tete cross.
Sunderland thought they had scored the winner when 15-year-old Christopher Rigg had the ball in the back of the net in the 90th minute, but the goal was rightly ruled out for offside to ensure a replay will be needed at the Stadium of Light.
'A strange game'
Fulham boss Marco Silva: "Strange game I can say. It was an intense game, a typical FA Cup match I think that's fair to say.
"Of course, we were more dominant but the game was too broken for me as a manager to enjoy it, to be honest.
"We conceded a cheap goal but, overall in the game, I think we've created enough chances to score more goals. On the other side I think we gave many opportunities for them to score as well and some counter-attacks where we gave the ball away in certain moments.
"We lost some important challenges and that was the part of the game I didn't like."
Mowbray proud of young Black Cats
Sunderland head coach Tony Mowbray: "The overriding thing today was the positivity of an extraordinarily young team that was on that pitch.
"Once Ross came off there were only two players over the age of 22 or 23 on the field and they played against Fulham, who beat Chelsea out here a couple of weeks ago and are riding high in the Premier League.
"So I think it was a good performance from a very young team that played with a good attitude and who threatened their goal from the start and could have added to our scoreline."
What's next?
Fulham return to Premier League action on Friday, with a trip to west London rivals Chelsea, live on Sky Sports; kick-off 8pm.
Marco Silva's side then host Nottingham Forest on Saturday February 11; kick-off 3pm.
Sunderland's next game is in the Championship, with a trip to Millwall on Saturday; kick-off 3pm.
Tony Mowbray's outfit then host Reading on Saturday February 11; kick-off 3pm.