Assombalonga's predatory flick in the second minute proved enough to ensure Boro made it three wins in a row for the first time since August and keep hopes of promotion alive.
Boro boss Tony Pulis' former club kept plugging away in search of an equaliser but found no way through to leave the Teesside crowd delighted.
Stoke caused problems but they are already planning for next season under Nathan Jones.
For Boro – in the midst of a defensive crisis following injuries to George Friend, Daniel Ayala and Dael Fry – Bristol City's draw with Reading earlier in the day meant a win would mean climbing back into a play-off spot.
And it took less than two minutes for the hosts to take advantage.
Assombalonga timed his run into the box perfectly to flick in Jonny Howson's cross after John Obi Mikel's excellent diagonal pass into the right corner proved perfect for the wing-back to run on to.
As on several occasions this season, Boro were unable to add to their opener and Stoke threatened a few times to level before the break, even if goalkeeper Darren Randolph was not posed too many serious questions.
The Potters enjoyed plenty of the ball. Boro's Aden Flint had to be alert to head clear when Tom Edwards' deep cross – after George Saville had been caught out – looked like it could be turned in by Sam Vokes from close range.
Vokes' strike partner Benik Afobe was picked out moments later when his six-yard header dropped inches wide of Randolph's left upright from James McClean's ideal delivery from the wing.
Another McClean cross was headed wide by Vokes just short of the break, providing another reminder that Boro had plenty work still to do.
And, after Ryan Shotton had headed over for the home side, Afobe glanced another McClean cross – this time from a corner – wide of the target at the opposite end early in the second half.
Middlesbrough, who introduced Stewart Downing on the left for the final 35 minutes, picked things up briefly after that. Saville and Ashley Fletcher had efforts on goal, even if goalkeeper Adam Federici found them routine.
The hosts then withdrew and invited Stoke to push on for much of the closing stages, but Boro's defence held firm.