The game swung on the two goals which arrived in the space of two minutes.
Carlow started impressively but Wexford scored six unanswered points in a 15-minute spell to lead 0-7 to 0-3 after 20 minutes.
When it looked as if Wexford were about to take complete control, it was Carlow who responded with quick goals from Shane Webb and Ronan O’Brien for a 2-6 to 0-7 interval lead.
Carlow extended that out further in the second period and coasted home to book their place in the quarter-final against Westmeath.
Scorers for Carlow: N Quinlan 0-8 (5f, 1 45); S Webb, R O’Brien 1-0 each; F O’Toole, J Dwyer, C Blake 0-1 each.
Scorers for Wexford: D Shanley 0-4 (4f); P Conway, K O’Connor 0-2 each; E O’Byrne 0-1.
CARLOW: A Murphy; K Chattan, D Hickey, G Power; F O’Toole, N Swan, C Murphy; J Dwyer, R O’Brien; S Webb, M Fury, S McGrath; N Quinlan, A Beaton, C Blake.
Subs: T Kenny for Fury; D Walsh for Webb; C Fenlon for Beaton.
WEXFORD: M O’Toole; T McDonald, P Cody, S Donohoe; G Staples, J Tubritt, S Byrne; P Barden, E O’Byrne; A Butler, D Shanley, D Doyle; P Conway, K O’Connor, P Breen.
Subs: C Devereux for Donohoe; N Rossiter for E O’Byrne; C Cleary for McDonald; P Shannon for Staples.
Referee: B Cawley (Kildare).
By Padraig O’Brien
Longford overcame the handicap of having defender Jason Earley sent off midway through the first half in eventually getting the better of Laois in the Leinster MFC qualifier at Glennon Brothers Pearse Park last night.
The sides were deadlocked at 0-5 apiece at the break but the home county made a great start to the second half to register 1-2 without reply, with corner-forward Larry Moran firing home the crucial goal.
Laois came storming back to score four points but their hopes were eventually dashed in the 50th minute when Longford substitute Liam Connerton grabbed a cracking goal.
Laois were thrown a late lifeline with a goal direct from a Paul Kingston free in the 58th minute but Longford held out for a well deserved win to earn a quarter-final clash against Dublin at Parnell Park on May 19.
Scorers for Longford: L Moran 1-3, R Connor 0-4 (4f), L Connerton 1-0, R McNerney, D McGivney, A McElligott 0-1 each.
Scorers for Laois: P Kingston 1-5 (1-5f), E O’Carroll 0-3, E Lowry (1f), D Cullen 0-1 each.
LONGFORD: C Farrell; M Mulligan, C Farrelly, J Gill; S Clarke, J Earley, L Sullivan; A McElligott, T Jackson; C Mallon, D McGivney, A Dalton; L Moran, R Connor, R McNerney.
Subs: L Connerton for T Jackson (ht); K Farrell for C Mallon (43); G Connell for L Sullivan (47); B Walsh for D McGivney (55).
LAOIS: A O’Reilly; J Bartley, C Dunne, R Fennell; M Scully, C McEvoy, R Dowling; J Nerney, D Mahon; N Mullins, P Kingston, D O’Reilly; E Lowry, E O’Carroll, D Cullen.
Subs: M Lyons for D O’Reilly (23) ; N Murphy for R Fennell (35); C Hogan for R Dowling (35); J Moore for E Lowry (41); N Brennan for N Mullins (54).
Referee: D Fedigan (Louth).