Robson scored a sweet left-footed volley to give Boro a first-half lead, but they had to settle for a replay after the Scotland midfielder gave away possession and allowed Fraizer Campbell to fire home an equaliser after the break.
"That was a bit of slackness from myself and I'm disappointed," Robson told ITV of losing possession.
"It was a simple little task, but it happens, top footballers do it so we'll move on."
And Robson was adamant Boro would back themselves at home, having matched Sunderland for long stretches.
"I think there are a few players in our team who have played at a higher level. We know what to expect, we're used to it. Credit to them, but we look forward to the next game and we think we can beat them playing at home."
Campbell was making his first senior appearance since August 2010 after a long-running injury nightmare, and was delighted to get straight back into the goals.
"Personally, I was delighted to be back out there," he said.
"We started sloppy, and let them dictate to us, but we dug deep. Cup games are like that, especially a local derby which adds that little bit more spice, but I think we deserved the draw and we move on to the next game."
Reflecting on his long spell on the sidelines, Campbell added: "You can't put it into words. It was a tough 18 months for myself, but days like this are what I've been waiting for, and hopefully now I can stay injury-free and just do what I do."
Boro boss Tony Mowbray admitted he was dealing with a "disappointed" dressing room after the match.
"In the second half a few things fell for us, and a clinical finish or ricochet could have got us the goal," he said.
"In the first half-hour, it was a bit attritional like a derby game can be, but we got the goal on the break, and maybe should have had a second on half-time. But great credit to the players as in the second half they got stronger."
Sunderland had a goal ruled out before the break, apparently for a mistaken offside decision, although two wrongs perhaps made a right as Craig Gardner had handled the ball before firing home.
Mowbray said he had not seen the incident, but questioned whether it may have played a role in other decisions in the match.
"I haven't seen it, but what I do think is the officials saw it," he said. "There seemed to be a lot of close decisions in the second half, and a lot of decisions went against us, maybe on the evidence of something they felt they got wrong.
"But I don't know. I haven't seen it. What I do know is that the linesman stood there with his flag raised for a long time."
Mowbray then called on Boro fans to turn out in force to give them home advantage in the replay.
"We've been better on the road than at home this year," he said.
"We've found it difficult to win at the Riverside but hopefully with a good atmosphere in a derby, and knowing what will be on offer in the next round, we can make it a great occasion."