With Middlesbrough defenders Emanuel Pogatetz and Andrew Taylor injured, 18-year-old defender Seb Caines scored in his club debut in the 32nd minute.
Mark Viduka added his first goal in the 49th.
Aiyegbeni Yakubu converted a penalty to make it 3-0 in the 57th.
Dawson began Hull's rally with a header in the 59th and Jon Parkin converted a 63rd-minute penalty to cut Boro's lead to 3-2.
Viduka scored his second goal in the 64th minute to make it 4-2, before Hull's Andy Dawson pushed it to 4-3 in the 69th minute with his second goal of the match.
Dawson almost completed a hat trick in the 80th minute, but Middlesbrough goalkeeper Mark Schwarzer tipped his curling free kick over the bar.
Manchester City win
Middlesbrough's Viduka battles for the
ball with Ian Ashbee of Hull [GALLO/GETTY]
Manchester City beat Sheffield Wednesday 2-1, with Darius Vassell scoring the winner for City in the 56th minute, shooting into the top corner after taking a pass from Bernardo Corradi.
Stephen Ireland had put Man City ahead in the 44th minute, before Lee Bullen leveled for Sheffield Wednesday in the 51st.
"We haven't fired on all cylinders and missed a lot of chances, but we are through to the next round and that is what matters in the FA Cup," City midfielder Joey Barton said.
"The onus is on the Premier League side to make sure there is not an upset. They took us all the way in the first leg. All credit to them."
City drew 1-1 at Sheffield Wednesday on January 7 and will play Southampton in the next round.
Coming up
Middlesbrough will face Bristol City in the fourth round, Barnet are to play Plymouth with Ipswich v Swansea and Wolves v West Bromwich Albion.
Luton's replay against Queens Park Rangers was called off because of a waterlogged field.
The two drew 2-2 at QPR on January 6.
Three third-round replays scheduled for Wednesday: Fulham v Leicester; Tottenham v Cardiff City; and Newcastle v Birmingham. Southend will play the winner of Tottenham-Cardiff match in the fourth round, set for January 27-28.