FA Cup - Campbell scores on Sunderland return
The forward had not played since August 2010 due to a persistent knee injury, but took just 13 minutes to find the net after coming on as a half-time substitute.
The replay will take place at Middlesbrough's Riverside Stadium on Wednesday, February 8.
Three minutes later, Boro went in front thanks to a moment of brilliance from Robson. Sunderland half-cleared the ball from their box, but Robson returned it with interest with a scorching low, left-foot volley that whistled past Mignolet on its way to the right corner of the net.
Winger James McClean had two chances to level - first when he cut in from the left and hit a low shot that Coyne blocked, then when he headed a cross straight at the goalkeeper.
Sunderland might have been level on 39 minutes when Craig Gardner fired in off the post, but was denied by an offside flag against Connor Wickham.
Replays showed Wickham was not directly interfering with play, but Gardner handballed in the build-up. Rightly disallowed, but for completely the wrong reason.
Having produced a moment of brilliance to give Boro the lead, Robson was guilty for facilitating Sunderland's equaliser.
A slack pass allowed McClean to race clear and square for Campbell, who kept his head to execute a cool finish.
Boro came back, though, and had chances to re-take the lead through Rhys Williams and Marvin Emnes.
Williams could have seen red for a reckless challenge on David Vaughan, but referee Kevin Friend thought a yellow card was enough.
McClean missed a late chance to nick it for Sunderland, but he hit his left-foot shot into the side-netting at the near post.