A Helder Costa volley and a close-range header from Ivan Cavaleiro put Wolves 2-0 up at the break and they looked comfortably in charge.
But Ruben Neves and then Matt Doherty were both dismissed after wild challenges earned them second yellow cards and left the visitors to play the last 20 minutes with a two-man disadvantage.
Middlesbrough laboured in possession but finally broke through, four minutes into added time, with a superbly taken strike from Patrick Bamford and then Stewart Downing fired wide in the final seconds.
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"There has been a lot gone wrong there and for the boys to do what they've done and dig in is credit to everyone," said Wolves skipper Conor Coady.
"We've done the same all season and we've got to keep on going. This is huge for us".
Second-placed Cardiff show no signs of offering hope to Fulham or the rest of the chasing pack and they made eight wins in a row with a 3-1 win over Burton Albion.
Kenneth Zohore opened the scoring for Cardiff, only for former England striker Darren Bent to level for relegation- threatened Burton.
But Nathaniel Mendez-Laing restored Cardiff's lead just before the break and Callum Paterson's close-range finish in the 64th minute completed the win for Neil Warnock's team.
Fulham went ahead at Carrow Road in the 66th minute thanks to a Stefan Johansen tap-in after Aleksandar Mitrovic's header was pushed out by Norwich keeper Angus Gunn.
Tom Cairney made it 2-0 four minutes later with a confident side-foot finish.
Millwall moved within a point of the playoff spots after an impressive 2-0 home win over Nottingham Forest.
Shaun Williams put the Lions ahead after 27 seconds with a header from a Ben Marshall cross and Lee Gregory added the second in the 33rd minute.
Bristol City, also in the push for the playoffs, drew 2-2 at lowly Barnsley thanks to a stoppage-time header from Josh Brownhill.
Fifth-placed Derby County were facing bottom club Sunderland later on Friday.
(Reporting by Simon Evans, editing by Ed Osmond and Pritha Sarkar)
