The pressure has been growing on Boro boss Southgate after a run of ten league games without a win had seen his side slip into the Premier League relegation zone.
But substitute Emnes struck in the 83rd minute to finally end the challenge of a dogged Wolves side.
Afonso Alves had fired Middlesbrough into a deserved half-time lead with a classy 44th-minute strike.
But their failure to turn that domination into a bigger lead looked set to haunt them when Sam Vokes grabbed a 63rd-minute equaliser with his first goal at Molineux.
Fortunately for Southgate, Emnes spared Middlesbrough's blushes when he converted a pass from skipper Stewart Downing.
But three minutes before that, Wolves should have taken charge of affairs when Andy Keogh picked out Matt Jarvis with a right-wing cross only for the former Gillingham winger to fire wide in front of an open goal.
After their recent poor run, it was no surprise that Middlesbrough made something of a nervy start that Wolves came close to exploiting.
Jason Shackell headed just wide before Matt Hill's powerful sixth-minute drive was bravely blocked by David Wheater.
Having managed to overcome that early storm, Middlesbrough were able to impose themselves on the game and it was something of a surprise that it took them until the stroke of half-time to break the deadlock.
The impressive Adam Johnson forced goalkeeper Wayne Hennessey into a smart 19th-minute save after good work from Alves.
Johnson was then just off target when he met a cross from Downing before the England Under-21 international was denied by a goal-line clearance from Dave Edwards.
Downing again picked out Johnson in the 37th minute and he walked the ball around Hennessey only to see a diving Edwards turn away his goalbound shot.
Wolves' luck couldn't last forever and it ran out 60 seconds before half-time when their defence was again split by a pin-point pass.
This time it was Tuncay who was the provider with a chip into the path of Alves that found the former Heerenveen forward unmarked and he was able to chip the ball over the advancing Hennessey from ten yards.
Alves should have made the game safe seven minutes into the second half only to see his rising drive pushed around the post by Hennessey.
It was a crucial save by the Welshman as it sparked Wolves back into life and they were rewarded with an equaliser 11 minutes later.
Middlesbrough made the mistake of giving Jarvis room on the right wing and, when he lifted a centre into the area, Vokes rose to power home a header from 12 yards past the stranded Brad Jones.
Ironically, after being pegged back, Middlesbrough stepped up their game and Johnson was denied a deserved goal when his powerful drive was well held by Hennessey.
Jarvis then wasted his golden opening and Emnes took full advantage to send Wolves crashing to only their third home defeat of the season.