Blues were the better side for large periods of the game and took the lead in the 55th minute when Aaron Pierre powered home a header from Sam Wood’s well-worked short corner.
The visitors responded almost immediately when Kyle Dempsey converted Stephen Elliott’s cross, but Wanderers were not to be denied and Hayes converted from the spot in the 75th minute after Sean O’Hanlon handled in the area.
Two became three in the 80th minute when Murphy fired home at the far post for his second goal in as many League 2 games for the club.
Carlisle’s misery was compounded two minutes later when Alex Marrow was sent off after picking up his second booking for a rash challenge on Max Kretzschmar.
Gareth Ainsworth named an unchanged side for the third successive game as Sido Jombati, Joe Jacobson, Alfie Mawson and Peter Murphy made their home debuts. Fellow new recruit Paul Hayes again skippered the Chairboys.
Wanderers started brightly and Max Kretzschmar saw a free-kick blocked before seeing two penalty appeals turned down after going down in the area.
Murphy was the next to try his luck but his ambitious first-time volley from the edge of the area flew wide in the eighth minute. The central midfielder was beginning to see more of the ball as moments earlier his exquisite outside-of-the-foot pass released Sam Wood on the left, but the winger’s low cross was well cleared.
Carlisle began to grow in stature and Brad Potts had the visitors’ best early chance but Matt Ingram got a touch on his effort to divert the ball wide after a mix-up in Chairboys’ defence.
Murphy thought he had come within inches of giving Ainsworth’s men the lead in the 21st minute as his header from Paris Cowan-Hall’s cross crashed back off the crossbar, but the assistant referee had already raised his flag.
Hayes fired over from distance following Matt McClure’s lay-off as Wanderers continued to look more of an attacking threat. Cowan-Hall was next to shoot goalwards, but his volley from inside the area flew harmlessly over the crossbar.
McClure drilled wide of Mark Gillespie’s right-hand post, but the best opportunity of the first half came eight minutes before the break and the way of Wycombe. Matt Ingram’s long clearance eluded the Carlisle centre-backs and Cowan-Hall’s superb touch took him through on goal, however, he elected to shoot early and his 20-yard strike cleared the bar.
Hayes called Gillespie into action in the 41st minute as he collected Jacobson’s long ball before turning his man and curling goalwards, only for Carlisle’s No.1 to make a smart stop low to his left.
Half-time: Wanderers 0-0 Carlisle
Blues were fast out of the traps after the break and Cowan-Hall acrobatically fired off target before Wood blazed wide after clever link-up play with McClure.
Carlisle substitute Pat Brough came to his side’s rescue five minutes after the restart when he hooked away from near his own goal-line after McClure had tried to divert home Wood’s low cross.
Chairboys didn't have to wait much longer for a deserved opener though as it came in the 55th minute. Wood played a short corner to Hayes before collecting the rebound and sending an inch-perfect cross to the back post which Aaron Pierre powerful headed home from six yards.
The lead was short-lived though as parity was restored little over 60 seconds later when Stephen Elliott’s cross was clinically turned home by Brad Potts.
The equaliser sparked the visitors into life and Kyle Dempsey turned inside Jombati before testing Ingram low down.
Kretzschmar fired a long-range free-kick over the bar as Wanderers began to look increasingly frustrated, but their persistence paid off 15 minutes from time when Sean O’Hanlon handled
Ingram’s free-kick in the area. Skipper Hayes stepped up and coolly sent Gillespie the wrong way from the spot to restore Blues’ advantage.
Ainsworth’s men deservedly gave themselves some breathing space five minutes later when Murphy stabbed home at the far post for his second goal in the famous quarters after Wood had just been unable to get on the end of a deep cross from the right.
Things went from bad to worse for Carlisle as Alex Marrow received his marching orders with eight minutes remaining after picking up his second yellow card for a crude challenge on Kretzschmar.
Aaron Holloway was introduced alongside Danny Rowe a minute later for his home debut and Blues comfortably saw out time with the visitors not threatening Ingram’s goal.
Wanderers: Ingram, Jombati, Jacobson, Pierre, Mawson, Cowan-Hall (Rowe 84), Murphy, Kretzschmar, Wood, Hayes (c) (Craig 81), McClure (Holloway 84).
Unused subs: Richardson, Bloomfield, Walker, Morias.
Carlisle: Gillespie, Symington, Robson (Brough 46), Meppen-Walter, O’Hanlon, Sweeney, Marrow, Dempsey, Potts, Dicker (Elliot 46), Paynter (Kearns 67).
Subs: Hanford, Grainger, Thirlwell, Gillies.
Attendance: 3121 (310 Carlisle)
50/50 winning number: 105576 (Prize fund £528)