Fortune, who only arrived on loan from AS Nancy on Thursday, grabbed Albion's second goal in the 54th minute as they recorded their first-ever double in the Premier League.
Chris Brunt had given Tony Mowbray's side a dream start with a fourth minute opener and Robert Koren set the seal on their win as Albion secured a third successive home victory - something they had not done in the top-flight for 24 years.
The latest success gave their survival hopes a fresh boost as they climbed off the bottom of the table and moved them level on points with a Boro side who are in free-fall.
Gareth Southgate's team have failed to win for ten league games, but could have taken something from the match had Alfonso Alves been more clinical in front of goal in the first half.
They were already 2-0 down when they were reduced to ten-men by the 62nd minute dismissal of Didier Digard for a lunge on Borja Valero and that ended their challenge once and for all.
Brunt capitalised on a mistake by the Middlesbrough right-back Tony McMahon to hand Albion into the lead.
When Carl Hoefkens sent his cross into the Boro box, McMahon's weak header flew straight to Brunt.
Brunt let fly with a left-foot drive that was deflected off McMahon and past goalkeeper Ross Turnbull.
The visitors launched a determined search for an equaliser through Alves, but the Brazilian-born marksman was out of luck in front of goal.
The former Heerenveen striker headed on to the bar just 60 seconds after Middlesbrough fell behind and then fired wide from close range when he latched on to a David Wheater free-kick.
Albion should have taken full advantage of the failings of Alves to increase their lead before half-time, but Jay Simpson was denied by Turnbull when he only had the keeper to beat before Brunt headed wide.
Turnbull then turned away a 37th minute shot at his near post from Fortune.
Middlesbrough raced straight up the field after that let-off but again Alves was frustrated when he shot straight at Scott Carson from 12 yards.
That was a crucial miss as Albion made the game safe through Fortune nine minutes into the second half.
Koren fed the ball in to Fortune who played a pass back to the Slovenian midfielder. Koren then let fly from the edge of the area with a shot that Fortune deflected past Turnbull from ten yards.
That was the end for Middlesbrough - especially as just eight minutes later Digard saw red.
Albion quickly took advantage of their greater manpower when Brunt played a sweeping crossfield ball to Koren, who hit a low drive beyond Turnbull from the edge of the area.