Craig opened the scoring from the spot three minutes into the second period after being brought down by keeper Ben Garratt, before Doherty sealed a second-round tie at home to Preston with Blues’ second in the 68th minute.
Chairboys were never threatened and Matt Ingram was barely tested throughout the 90 minutes as Gareth Ainsworth’s side banished memories of last year’s heavy defeat at Gresty Road. There was a further milestone with Matt Bloomfield coming on for his 300th appearance for the club.
Gareth Ainsworth made three changes to his side with Gary Doherty, Matt McClure and Steven Craig replacing Leon Johnson, Max Kretzschmar and Dean Morgan.
The first half chance fell to the Blues within the opening 60 seconds, but Josh Scowen didn’t connect cleanly with an attempted 20-yard volley and the ball rolled through to keeper Ben Garratt.
Billy Knott provided the first real moment of quality and it came in the sixth minute when the Sunderland loanee picked out Paris Cowan-Hall in acres of space on the edge of the area, but the winger shot first time and the ball flew wide.
Matt McClure dragged a free-kick wide before Knott fired over with an ambitious volley from the edge of the area, but the Wanderers had started well and were the better side in the opening 10 minutes.
McClure had the first clear-cut chance in the 17th minute when a neat passing move involving Scowen, Knott and McCoy saw the striker receive the ball in space in the area, but he chose to shoot first time and his strike was straight at Garratt.
Good play from Craig and Knott on the left saw the latter find space, and he picked out McClure in the area, but the forward’s shot deflected wide.
Crewe struggled to deal with the resulting set-piece, but both Craig and Doherty were unable to force the ball over the line.
The home side’s first effort came in the 27th minute when Clayton’s low strike was straight at Ingram, and that chance drew an ironic round of applause from the Crewe support, such was Blues’ positive start.
Crewe began to see more of the ball with Max Clayton and Ajay Leitch-Smith looking lively, however, it was Ainsworth’s men who came close to breaking the deadlock three minutes before the break.
Marvin McCoy did well to dispossess Byron Moore on halfway and the ball fell to Cowan-Hall, who drove forward before trying his luck from just inside the area. The shot was straight at Garratt, but the inexperienced keeper fumbled and he managed to recover just in time before McClure could reach the rebound.
Blues controlled large periods of the first half with Craig getting joy against the centre-backs, however, Wanderers found it tough creating clear-cut chances against their League 1 opponents.
Chairboys’ play was rewarded three minutes after the break when Garratt tripped Craig in the area after the striker had reacted quickest to the keeper spilling McCoy’s speculative strike. Craig took the spot-kick himself and sent the keeper the wrong way for his second goal in the famous quarters.
The hosts were forced into their first change in the 56th minute when Clayton was forced off following a crunching challenge from Scowen, and the Blues continued to push forward with Cowan-Hall flashing a header just wide on the hour.
Crewe boss Steve Davis made his second substitution in the 66th minute as Brad Inman replaced Moore, who had been well marshalled by Wood.
And Wycombe gave themselves breathing space two minutes later when McClure flicked on Scowen’s corner and Doherty slid in to fire low past Garratt. The goal was no more than they had deserved after completing dominating after the restart.
Garratt kept Crewe in the contest in the 77th minute when the ball broke to Cowan-Hall on the edge of the area and his powerful first-time effort was well stopped low down to the keeper’s right.
Ainsworth made his first change 10 minutes from time, bringing on Matt Bloomfield for Knott and the appearance marked the midfielder’s 300th for the club.
Inman finally called Ingram into action with two minutes remaining, but his low shot was straight at keeper, who then comfortably saved from Vadaine Oliver.
Junior Morias was introduced in injury-time as Craig went off to a standing ovation from the travelling faithful, and the Blues comfortably saw time out to seal a thoroughly-deserved victory and book a second-round tie against Preston North End on the weekend of 7/8th December.
Wanderers: Ingram, McCoy, Wood, Doherty, Stewart, Cowan-Hall, Lewis (c), Scowen, Knott (Bloomfield 80), Craig (Morias 88), McClure.
Subs: Harrison, Johnson, Spring, Kretzschmar, Morgan.
Crewe: Garratt, Mellor, Guthrie, Davis, Ellis, Osman (c), Evans, Grant, Moore (Inman 66), Clayton (Oliver 56), Leitch-Smith.
Subs: Martin, Audel, Waters, Nolan, Aneke.
Attendance: 1,695 (81 Wycombe)