Peter Odemwingie gave Albion an early lead only for Latics to hit back through Charles N’Zogbia and Ben Watson.
But Fortune salvaged a share of the spoils for the home side with 11 minutes remaining.
It keeps the Baggies three points ahead of Wigan with a game in hand.
Albion were again left to rue poor defending at set-pieces for both the goals they conceded and Latics posed a threat through N’Zogbia throughout.
But the Baggies also asked questions of the visitors’ defence through Odemwingie.
West Brom boss Roberto Di Matteo handed a debut to new loan signing Carlos Vela from Arsenal.
Wigan were the first to threaten and hesitancy by Paul Scharner almost let in Hugo Rodallega inside 30 seconds.
He allowed the ball to bounce in the Albion box and it needed an interception from Gonzalo Jara to deny the Wigan striker a clear shot at goal.
But after five minutes the Baggies went ahead through Odemwingie in their first meaningful attack.
Jerome Thomas was the creator with a perfectly-weighted ball which picked out the run of Odemwingie who beat the offside trap.
He kept his cool before sliding the ball past Ali Al Habsi into the corner of the net.
It was Odemwingie’s ninth goal of the campaign – all in the Premier League.
Albion were on top and Jonas Olsson sent a powerful header past the post from another Brunt free-kick.
But Wigan drew level in the 20th after Youssouf Mulumbu brought down N’Zogbia on the edge of the box.
N’Zogbia took the free-kick himself and it deflected off Mulumbu before flying past Boaz Myhill into the corner of the net.
It was N’Zogbia’s first goal in 13 matches.
N’Zogbia was proving a handful and one powerful run took him past Scharner and Jara before his volley from a narrow angle was beaten out by Myhill.
Brunt drilled a low drive just past the post after being teed up by Vela.
But Wigan took the lead in the 42nd minute after Jara was booked for chopping down Maynor Figueroa.
Watson drilled the free-kick into the danger area from the left of the box and no-one got a touch before it flew into the far corner of the net.
Albion made a positive start to the second period and Marek Cech was only just too high with a 25-yard drive.
But Myhill needed to react sharply to block a shot from Franco Di Santo.
The home side started to exert some pressure and Thomas curled a shot from inside the box just over the bar.
But Wigan were looking menacing on the break and Myhill blocked a shot from the dangerous N’Zogbia with his legs.
Albion built up a head of steam and Fortune crashed a 20-yard drive against a post.
The woodwork also denied Brunt after good play by Odemwingie.
With 11 minutes left Fortune brought Albion back on level terms.
Mulumbu sprayed the ball out to Thomas whose cross was headed home by Fortune.
Habsi produced a superb save to turn over Olsson’s header from Brunt’s cross.
Albion had penalty appeals rejected in injury-time when the ball bounced up and appeared to hit the hand of Gary Caldwell.
Then Habsi denied Fortune at the foot of a post.