The Bees enjoyed a rampant start to the game and led from the 3rd minute when John Akinde latched on to a long ball, and tapped home after rounding Matt Ingram.
They squandered chances to go further ahead, not helped by a succession of fine saves by Ingram, and the Blues were given a route back into the game when Paul Hayes levelled from the spot after Matt McClure was fouled.
Sam Togwell then saw red for Barnet just before the break for a foul on Matt Bloomfield, and the visitors took full advantage, securing their path to round two with further goals from Aaron Pierre and Sam Wood before Andy Yiadom was sent off on 90 minutes to deepen the hosts' misery.
Gareth Ainsworth was dealt a blow in midweek by the news that Brentford and Rotherham had denied Alfie Mawson and Danny Rowe permission to play in the FA Cup, which meant a reshuffle along the backline for the first time this season.
Sido Jombati moved to centre-back with Josh Scowen reverting to right-back as Max Kreztschmar came into midfield, while Matt McClure started up front in place of Steven Craig, who passed a late fitness test to make the bench.
After an impeccably observed minute’s silence for Remembrance Weekend, the game sprang to life inside three minutes when John Akinde beat the offside trap to meet a through-ball before rounding Matt Ingram and tapping home the opening goal.
It could have got worse on 14 minutes when Curtis Weston ran clear but saw his effort heroically saved by Ingram, who recovered quickly to grab the rebound with Akinde lurking.
Wanderers almost reacted in the perfect manner but Matt McClure was unable to find the power to convert a chance after being picked out on the edge of the box.
Josh Scowen received his fifth yellow card of the season midway through the half for a foul, meaning he’ll miss the visit of Burton Albion in nine days’ time, and moments later, Ingram was called upon to make another fine save, tipping Lee Cook’s wind-assisted free-kick over the bar.
Matt McClure latched onto Sam Wood’s driven through-ball on 32 minutes but after getting the first touch, he was clattered by the onrushing keeper only to see the referee wave the appeal away, infuriating the Wanderers’ coaching staff.
However, the referee made the right call on 34 minutes when McClure was tripped in the box, giving Paul Hayes the chance to level the scores. He duly obliged, scoring his eighth goal of the season even though Graham Stack went the right way.
After the poor start they made, Wanderers may have been delighted to go in level at the break and they had even more cause for optimism when Barnet were reduced to ten men in first half stoppage time. Sam Togwell lunged in on Matt Bloomfield and was rightly sent from the field, just moments before the whistle.
The second half began with all the action at one end, just like the first period, although this time it was Wycombe who were on top. Graham Stack did well to turn Matt McClure’s 20-yard drive over the bar, before Hayes headed wide from Jacobson’s cross.
The pressure continued, and paid off. After Joe Jacobson’s long-ranger was deflected out for a corner, the left-back’s delivery was perfect for Aaron Pierre to thump his header into the back of the net.
Pierre quickly came down from his high with a booking which rules him out of the Burton match, but as the rain began to pour down, the Blues remained in control with their one-goal and one-man advantage.
And the lead was stretched with 15 minutes remaining when Matt McClure remained onside to clip in a cross-shot which sailed over the head of Stack and was met by Sam Wood virtually on the line to tap home his fourth of the season.
Ingram still had to be alert to keep out a couple of awkward low shots which skipped up off the slippery surface, but the result never looked in doubt after Wood’s strike. Andy Yiadom saw red in the closing stages for a foul on Josh Scowen but there was no further drama and the Blues are in the hat for the second round.
Wycombe: Ingram, Scowen, Jacobson, Jombati, Pierre, Bloomfield (Lewis 80), Kretzschmar, Murphy, Wood, Hayes (Cowan-Hall 75), McClure (Ephraim 82).
Unused subs: Richardson, Craig.