Reds conceded twice in the last eight minutes before losing a crazy battle with Barnet.
The home side were on their way to a routine win with two goals from set-pieces either side of half-time by Bira Dembele.
Then Crawley hit back with two goals in three minutes by Mitch Hancox and Jimmy Smith to make it 2-2 with just 12 minutes to go.
However Barnet responded with two more goals netted by former Red John Akinde from the penalty spot and an unlucky own goal by Lewis Young.
Crawley made one change to the side which beat Dagenham 3-2 with Simon Walton returning from a one-match ban in place of Roarie Deacon.
Barnet also made one change as cenre back Michael Nelson was preferred to Bondz N’Gala.
In a slow starting game, Barnet’s Aaron McLean had the first meaningful attack , hitting the wall from a free-kick before Josh Clarke saw his follow-up shot deflected over the bar.
Crawley made much of the running in the first half hour, Jimmy Smith’s chip on goal not having enough power to trouble Ian Lawlor.
Rhys Murphy, playing his last game of his three-month loan from Oldham, had the ball in the net after 14 minutes but the goal was disallowed for offside.
Gwion Edwards and Smith saw shots deflected wide for a corner.
Reds captain Sonny Bradley rose well to meet Luke Rooney’s corner with a bullet-header which was tipped over the bar by Lawlor.
Hancox created a chance for Murphy with a breakaway attack but the Crawley topscorer could not find his touch.
Barnet’s Josh Clarke fired wide from the edge of the area as the home side began to look up for the fight.
Barnet made a substitution just before half-time when Nelson went off injured and was replaced by N’Gala.
Barnet then took the lead on 45 minutes when Dembele headed in Luke Gambin’s corner at the near post.
The same two players combined to increase Barnet’s lead just seven minutes after the start of the second half, again Dembele heading home a Gambin free-kick low into the bottom right corner.
Clarke blasted a chance low at Flahavan from the edge of the box, then at the other end, Matt Harrold headed a ball from Hancox narrowly wide.
Barnet continued to press forward as Curtis Weston had an effort tipped over for a corner by Flahavan.
Just as it seemed like Barnet were cruising to victory, Crawley managed to pull back to 2-2 with 12 minutes to go with two unexpected goals.
Hancox sent a cross into the area which drifted into the net on 75 minutes, much to the delight of the 300-strong travelling fans.
The supporters went wild when Reds scored again just three minutes later with a fantastic strike put away by Smith.
However Barnet earned a penalty for a foul on John Akinde, he took his time before placing the ball to the right of Flahavan.
Barnet then sealed victory with a fourth goal moments later, an own goal by Lewis Young who got in the way of a clearance by Walton which cannoned into his own net.
REDS: Flahavan, Walton, Bradley (capt), Edwards, Smith, Rooney (Deacon 65, Young, Harrold (Fenelon 65), Hancox, Yorwerth, Murphy
Unused subs: Barnard, Preston, Ashton, Jenkins, Emmanuel
BEES: Lawlor, Johnson, Dembele, Nelson (N’Gala 44), Yiadom, Weston (Champion 83), Akinde, Gambin, Togwell, McLean (Gash 65), Clarke
Unused subs: Lisbie, Batt, McKenzie-Lyle, Stacey
REFEREE: Iain Williamson
ATTENDANCE: 1,888 (323 away)
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