The hosts raced 1-2 to 0-1 clear following a 13th minute goal from PJ Banville.
Points from Raymond Galligan and Eugene Keating kept Cavan in touch before Ciarán Lyng converted a Wexford penalty. Cavan keeper James Reilly then dropped a high Ben Brosnan effort and Eric Bradley tapped the ball into an empty net.
But efforts from Galligan, Niall McDermott and Brendan Fitzpatrick left two between them, 3-2 to 0-9, at the break.
Galligan’s 51st minute penalty was pushed over the bar by goalkeeper Anthony Masterson to leave three points between the sides, 3-6 to 0-11. The incident also resulted in the dismissal of Wexford midfielder Daithi Waters.
Five minutes later a superb goal from midfielder Gearoid McKiernan left just a point between the sides. However, the Ulster side failed to build on that effort as Brosnan pointed a free before Eric Bradley beat keeper Reilly from close range to open a decisive 4-7 to 1-11 lead.
Scorers for Wexford: E Bradley 2-0; C Lyng (1-0 pen), PJ. Banville 1-1 each; B Brosnan 0-5, 3f, 1, 45; A Flynn, P Byrne, 0-1 each.
Scorers for Cavan: R Galligan 0-5, 3f, 1pen; G McKiernan 1-0; N McDermott (0-3, 1f; N Smith 0-2; B Fitzpatrick, E Keating, D McKiernan 0-1 each.
WEXFORD: A Masterson; N Murphy, G Molloy, B Malone; L Chin, A Flynn, A Doyle; D Waters, A Shore; E Bradley, R Barry, B Brosnan; P J Banville, C Lyng, P Byrne.
Subs: C Morris for Banville; D Murphy for Chin; R Quinlivan for Shore; J Holmes for Barry.
CAVAN: J Reilly; D Barkey, S McCormack, R Flanagan; D O’Dowd, P O’Reilly, B Waters; D Givney, G McKiernan; N Smith, N McDermott, S Jordan; B Fitzpatrick, E Keating, R Galligan.
Subs: K Fannin for Jordan; J McCutcheon for Waters; M Dunne for Galligan; D McKiernan for Keating.
Referee: M Collins (Cork).