Boro brought an end to a run of four consecutive home defeats by putting a dent in Hull's own attempts to make a late climb into the top six.
Assombalonga was in the right place to turn in Ashley Fletcher's low cross with 25 minutes on the clock and that was how it stayed at a nervy Riverside – despite Kamil Grosicki hitting the bar with a late free-kick.
Hull, who had won their previous three games, had a couple of half-chances to level things but that never materialised and Middlesbrough can still dream of a Premier League return.
Middlesbrough, who had trailed the play-off zone by four points before the game, were keen to avoid a fifth straight reversal on home soil for the first time since 1996 – and to do that Tony Pulis kept the same team that beat Bolton in midweek.
Hull looked the more confident of the teams at the start and enjoyed far too much freedom down the flanks and Grosicki and Jarrod Bowen looked to make the most of the space. Grosicki had the best early chance when he rolled an effort to goalkeeper Darren Randolph.
But after failing to capitalise, Middlesbrough settled themselves and, after a change of formation from Pulis, they finished the half the stronger and created the best chances.
Defender Aden Flint headed wide from Jonny Howson's corner seconds before the opening goal. Ryan Shotton played Fletcher in down the right and his delivery was turned over the line from close range by Assombalonga.
By the time the half-time whistle had blown, Middlesbrough were left to wonder how they were not more than a goal to the good.
Howson and George Saville both got in each other's way at the back post when it looked like one would turn in a knock-down from Fletcher minutes later, while the latter somehow volleyed over a deep Shotton cross when he was left totally unmarked 10 yards out.
After the break Hull kept pressing in search of an equaliser and that opened things up more. They were left puzzled when Fraizer Campbell somehow headed wide from close range at the back post after the hour.
It seemed a matter of time before Hull would equalise but Middlesbrough, who went close to extending their lead when Fletcher was denied by goalkeeper David Marshall, kept clearing their lines.
Former Chelsea midfielder John Obi Mikel, the Middlesbrough captain, headed off the line after a period of pinball in the home penalty area, while Assombalonga ought to have made it two when he messed up a pass rather than shooting when faced with just Marshall to beat.
Campbell had a strong penalty shout waved away before Grosicki's late free-kick hit the underside of the bar, bounced on the line and Flint cleared.