Boro old boy Reach’s excellent goal in the 20th minute proved to be the difference for the Owls, who dug in again following the departure of Jos Luhukay last week.
Wednesday might be up for sale and searching for a new manager, but it was the Middlesbrough fans left most disappointed and annoyed on Boxing Day.
This was the seventh time this season Middlesbrough have not won on home soil, leaving them nine points behind the top two a year on from Tony Pulis taking over on Teesside.
Boro hoped to deliver a second December win in a row after a disappointing spell before Christmas.
Wednesday had seen signs of improvement over the weekend when they beat Preston under the guidance of Bullen after Luhukay left.
Middlesbrough, who had five at the back but asked wing-backs Paddy McNair and George Friend to attack, made the strongest start to get the Owls on the back foot.
Stewart Downing, operating as a number 10 after his recall to the team, was quick to force Sheffield Wednesday goalkeeper Keiren Westwood into a low save at his near post and that led to some decent pressure.
But the brief positivity soon turned in Sheffield Wednesday’s favour when Barry Bannan started to find more freedom to cause problems.
And it was the Scotland international’s clever pass in behind the Middlesbrough defence that set Reach free. Reach then took it down brilliantly before he applied the low finish beyond goalkeeper Darren Randolph.
The home fans booed and sarcastic chants of “attack, attack, attack” could occasionally be heard knowing Middlesbrough have lacked goals and quality in the final third this season.
Sheffield Wednesday looked confident and capable of holding on at the break. Bannan dictated things in the middle, with Reach dangerous when he got on the ball. Westwood was not seriously tested.
Middlesbrough changed system for the second half and striker Jordan Hugill was introduced for defender Dael Fry.
Even then it was a real struggle to test Westwood, with the best they could serve up a wayward McNair effort from distance.
Sheffield Wednesday stayed strong. In fact they almost grabbed a second on the hour when former Sunderland striker Steven Fletcher almost got on the end of another clever pass from Bannan.
Marcus Tavernier’s attacking intent was given the final half hour and soon after that Middlesbrough finally recorded a second effort on target.
Daniel Ayala’s header from McNair’s corner was comfortably held by Westwood, shortly before Tavernier’s flick had to be saved by the keeper too.
Lewis Wing drilled an effort wide, but this was Wednesday’s festive party.