Two second half John Akinde penalties either side of a Harry McKirdy header gave the Bees all three points in difficult conditions.
On a wet afternoon at The Hive London, both sides took their time to adjust to the conditions. In the 12th minute, Akinola broke well down the left wing before delivering a low cross that was just behind the onrushing Bees players.
Curtis Weston then powered is way into the area but was cut out by Ng, the ball fell to Akinola but the wide man curled an effort just around the far upright from the edge of the box.
Shortly before the 30-minute mark, Ryan Watson skimmed an effort off the surface testing the Crewe goalkeeper Ben Garratt.
The Bees certainly edged the first half and had the best chance when John Akinde played an excellent ball through for Alex Nicholls to get on the end of, he cut inside before shooting past the right post.
Crewe would have two more efforts before the break, but Jones fired well over the crossbar and Ainley hit a first-time volley straight at Craig Ross.
Barnet were awarded a golden opportunity to take the lead 5 minutes after the restarted when referee Lee Swabey was given no choice other than to point to the spot when Akinola was brought down by Jamie Jones.
John Akinde stepped up and confidently rolled it past Garratt, sending the Crewe 'keeper the wrong way and giving the Bees a 1-0 lead.
Craig Ross was forced into an excellent save in the 67th minute when Paul Green's curling effort looked destined to end up in the net, but Ross flicked it around the post with a fingertip save.
Crewe got an equaliser in the 70th minute when the Bees couldn't clear a free kick, George Ray looped a header towards the far post for McKirdy to head home.
However, this setback spurred the Bees on to go again and look for another to retake the lead. Substitute Jean-Louis Akpa Akpro had a first-time volley smack the crossbar before Watson fired wide.
Coulthirst wriggled his way into a shooting position but had an effort well blocked and then, with four minutes to go, Barnet were awarded another penalty when George Ray brought down Coulthirst.
Akinde stepped up yet again and fired it high into the top right corner, a spot-kick that even that the very best 'keepers would have struggled to keep out.
The Bees then defended resolutely to earn three points and send the Barnet fans, who created a great atmosphere right throughout the game, home happy.
Attendance - 1,951 (379 away)
Man of the Match - Dan Sweeney
Barnet: Ross, Brindley, Santos, Nelson (Shomotun 77'), Watson, Weston, Akinde, Tutonda, Akinola (Coulthirst 58'), Nicholls (Akpa Akpro 74'), Sweeney.
Unused - K. McKenzie-Lyle, Clough, Fonguck, Payne.