Ian Byrne helped himself to 1-7 as Wexford got their Leinster SHC campaign off to a winning if unconvincing start in Mullingar.
The Slaneysiders will need a big improvement against holders and All-Ireland champions Kilkenny in a fortnight's time.
Westmeatht ook an hour to get going and it wasn't until a serious injury to free-taker Derek McNicholas late on that the hosts found their best form, sniping 1-4 in the closing stages to leave Wexford with some questions to answer.
McNicholas left the field on a stretcher after an apparent neck injury, necessitating 12 minutes of injury-time.
Liam Dunne's visitors barely needed to play above fourth gear for the first 60 minutes or so and were 15 points ahead at one stage.
Westmeath were also rocked before throw-in by the loss of Shane Power in the warm-up.
The Clonkill defender crumpled to the ground after being apparently struck on the face guard by a stray ball.
He received medical attention before being helped off the field clutching his face. With Gary Greville also ruled out, it meant Westmeath had two changes to their official full-back line.
Wexford: M Fanning; E Moore, M O'Hanlon, L Ryan; A Shore (0-1), L Chin, D O'Keeffe; D Redmond (0-3), H Kehoe (0-1); J Guiney (0-1), D Waters (0-2), I Byrne (1-7, 0-6f, 0-1 65); P Morris (0-5, 0-1f), C McDonald (0-3), L Og McGovern (0-1). Subs: K Foley for Guiney, R Jacob for McGovern, G Sinnott for Kehoe, S Tomkins for McDonald, P Doran for Byrne.
Westmeath: S McGovern; K Duncan, T Doyle, L Varley; A Craig, A Clarke (0-1), A McGrath; E Price, C Boyle (0-1); A Devine (0-1), D McNicholas (0-5, 0-5f), J Clarke (0-1); N Dowdall (0-2), B Murtagh (1-4, 0-1f), D Clinton. Subs: E Corrigan for Price, K Murphy (0-01) for McNicholas.
Ulster SFC quarter-final: Derry 0-12 Down 0-11.
Leinster SFC quarter-final: Laois 0-16 Kildare 0-16.
Ulster MFC quarter-final: Derry 2-11 Down 1-7.
Leinster SHC quarter-finals: Laois 0-29 Offaly 0-21; Wexford 1-24 Westmeath 1-16.
LEINSTER SHC quarter-final replay: Galway 5-19 Dublin 1-18.
Munster SHC semi-final: Waterford 3-19 Cork 1-21.
Christy Ring Cup final: Kerry 1-20 Derry 0-12.
Nicky Rackard Cup final: Roscommon 2-12 Armagh 1-14.
Lory Meagher Cup final: Fermanagh 3-16 Sligo 1-17.