The Owls are four points adrift of safety with two games remaining after they were beaten on Teesside, and must avoid defeat in their last two matches against Nottingham Forest and Derby to have any chance of avoiding the drop to League One.
They looked to be heading for a positive result when Josh Windass' penalty cancelled out Yannick Bolasie's opener at the end of the first half, but second-half goals from 18-year-old substitute Josh Coburn and Duncan Watmore enabled Middlesbrough to claim their second success in the space of four days.
Boro completed a quickfire South Yorkshire double after triumphing at Rotherham on Wednesday night, making the most of some poor Sheffield Wednesday defending to claim all three points.
Bolasie was a constant threat for the hosts as he drifted across the frontline, and while he failed to find the target with a low effort in the sixth minute, the Everton loanee claimed the opener 14 minutes later.
Julian Borner's poor first touch enabled Bolasie to win possession, and after advancing into the right-hand side of the area, the Boro forward drilled a low shot into the bottom corner via a slight deflection off Tom Lees.
George Saville headed over as the home side continued to press, but with their Championship status on the line, Wednesday's players gradually roused themselves as half-time approached.
Andre Green's deflected effort whistled just past the post, before Borner spurned an excellent opportunity when he hooked wide from eight yards out.
The Owls were the better side in the closing stages of the first half, and their superiority was rewarded as they equalised six minutes before the break.
Green was tripped by Marc Bola as he dribbled across the face of the 18-yard box, and with Barry Bannan having been removed from penalty duties after his miss against Bristol City last weekend, Windass stepped up to fire a low finish past Jordan Archer.
Windass should have claimed a second goal four minutes after the interval, but after breaking on to Green's square ball in the area, the striker side-footed wide from close to the penalty spot.
Adam Reach was denied by a well-timed challenge from Grant Hall as he broke into the box moments later, but while Wednesday dominated possession for most of the second half, they fell behind in the 73rd minute.
Coburn had only been on the field for eight minutes when his persistence in the Wednesday box won a corner. A short-corner routine resulted in Neeskens Kebano crossing from the right, and Coburn angled a header into the bottom corner to claim his first goal on only his second senior appearance.
Boro made the points safe from another corner with nine minutes left, with Watmore scrambling home from close range after Bolasie's header was blocked.