Lukas Jutkiewicz fired Boro ahead in the 83rd minute but Danny Hylton's 89th leveller appeared to have earned a replay until the Polish striker pounced again in stoppage-time.
Mowbray admits he is keen to avoid a fellow Championship team in the fifth round, but is not worried about facing top-flight opposition.
"Managers only ever want a home tie and in an ideal world it wouldn't be someone from our league," he said.
"I'd like to play a Premier League side or one of the lower-league teams who've done well today.
"The FA Cup starts getting exciting in the fifth round as you're one game away from the quarter-finals.
"Let's see who we get. If we play Manchester United or Arsenal the onus is not on us to break them down, it's on them to break us down and if they get frustrated every time a move breaks down we can counter.
"Either is suitable for us. A top team that we can frustrate or a lower league team we can give problems and try to get through to the next round."