Simon Donohoe netted for Wexford but impressive finishing from Sean Quigley and Tomas Corrigan gave the Erne side a 0-10 to 1-6 lead at half-time.
Fermanagh's Richard O'Callaghan was black-carded but the visitors surged clear midway through the second half at Wexford Park to secure victory.
Corrigan added six second-half points for a match total of nine.
The Ulster visitors enjoyed a wind advantage in the first half with Quigley and Corrigan each bagging two points to give Fermanagh a 0-4 to 0-1 lead.
Wexford stayed in touch courtesy of Donal Shanley, who scored their opening four points.
O'Callaghan hit the Erne net to put the hosts ahead for the first time but Fermanagh hit back with three of the final four points in the first half.
Wexford would have fancied their chances with the wind at their backs in the second half, and they quickly levelled with another Shanley score.
Fermanagh had penalty claims turned down early in the second half before the clinical Corrigan restored their lead.
Ciaran Flaherty proved an able replacement for the black-carded Callaghan as Pete McGrath's team rattled over five straight points.
Wexford gave themselves hope when points from John Turbitt and Shane Roche left just three between the sides.
Corrigan completed his tally for the game before an Aidan Breen point ensured Fermanagh's place in Monday's second-round draw.
Fermanagh have progressed to the All-Ireland qualifiers second round thanks to a 0-19 to 1-11 win over Wexford.