The Premier League side took a ninth-minute lead as Aaron McLean bravely stabbed home his first goal in 11 months.
A deserved second arrived 15 minutes into the second half as George Boyd gifted a goal to German striker Nick Proschwitz - a last-minute replacement for the injured Matty Fryatt in attack.
Victory was Hull's first in nine trips to Middlesbrough and moved the Yorkshire club into the fourth round for the third season in succession.
Both sides were much changed, with Aitor Karanka making six alterations to his Middlesbrough side and Hull boss Steve Bruce opting for nine.
The visitors' unfamiliar line-up showed no lack of cohesion though, as McLean raced onto David Meyler's deflected ball and beat back-up home goalkeeper Dimi Konstantopoulos with a sliding challenge.
Middlesbrough's attempts to get themselves back into the tie were characterised by wasteful possession.
As well as a lack of discipline on the ball, a number of fouls also gave the visitors opportunities to test Konstantopoulos, but the former Greece international dealt with several Boyd free-kick deliveries well.
Hull striker Gedo saw penalty appeals waved away while, at the other end, Emmanuel Ledesma's long-range effort was comfortably saved by Steve Harper as the visitors went in ahead at the interval.
Karanka made attacking substitutions at the break, introducing Albert Adomah and Lukas Jutkiewicz, and the Ghana winger was quickly into the action as he beat three challenges on the right - but his fellow substitute could not convert the resulting cross.
Maynor Figueroa delayed from Marvin Emnes' cross but Dean Whitehead could not capitalise, firing off target, and Hull ended the game as a contest just three minutes later.
Paul McShane released Boyd down the left and the Scotland international drove to the byline before cutting back to Proschwitz, who finished simply from six yards.
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