Coulibaly gave us a first half lead from the penalty spot after Gunners skipper Isaac Hayden was sent off for bringing down Ruben Lameiras, before Winks wrapped up a comfortable victory in the closing stages of the second period.
The headline team news was the inclusion of Vlad Chiriches at centre-half - the Romanian international skippering the side after recovering from a back problem that had kept him out since the end of January.
We made a bright start under the lights - and it was a trend that continued for the entire contest.
Gunners goalkeeper Matt Macey was a busy man from the outset, catching danergous balls into the area from both flanks - the first a header back across goal from Milos Veljkovic after Lameiras' corner and the second a Kenny McEvoy cross from the right.
Macey's opposite number Jordan Archer was required to make a low stop from Thomas Eisfeld in between that aerial double after Jon Toral's effort was partially blocked by Connor Ogilvie 20 yards out, but it proved the only real necessary participation for the Scotland Under-21 international in the first half as we pressed the Gunners back.
Our first clear-cut chance fell to Coulibaly after 12 minutes, but the striker dragged his shot wide after a neat interchange with Lameiras.
Macey gathered the ball again three minutes later after Nathan Oduwa's corner hit Coulibaly before Arsenal defender Semi Ajayi slid in to deny Oduwa as he tried to convert Lameiras' through pass.
Leander Siemann nudged the ball behind to prevent Alex McQueen's right wing cross from reaching the prowling Coulibaly, before our pressure finally told.
Ogilvie won a tackle in midfield and fed Lameiras who showed lightning-quick footwork to trick his way between Ajayi and Hayden - the latter tripping him as he went through.
Hayden was shown a straight red card, with Coulibaly blasting the resulting penalty high into the net as the game entered its 28th minute and the striker entered into orbit with an acrobatic celebratory somersault to the delight of the White Hart Lane faithful.
The visitors reorganised into a 4-4-1 formation to cope with their numerical disadvantage, but that didn't stop us going close to adding a second goal two minutes later as McQueen's low ball in from the right was deflected onto the post by retreating left-back Brandon Ormonde-Ottewill.
Dominic Ball - Vlad's partner at centre-half - showed his warewithal to cut out Eisfeld's attempted reverse pass to Austin Lipman, while the industrious Winks sent a couple of long-range shots over the bar as we closed out the first half in command.
We began the second period in the ascendancy too, Oduwa finding the side netting from a tight angle on the left and McEvoy being denied by Siemann before Lameiras curled a shot over the top.
Arsenal were presented with a chance to equalise when Eisfeld won a free-kick 20 yards out. The Arsenal man was back on his feet to address the set piece, but he curled it wide.
Lameiras and Coulibaly both tested Macey around the hour mark before Oduwa nearly doubled our lead. He latched onto the latest Lameiras through ball and, after seeing his initial shot blocked by Macey, he got a second bite of the cherry - only for Siemann to block.
On 62 minutes Lameiras showed great vision to play McEvoy in behind the defence but his shot somehow stayed out after hitting the sprawling stopper and the woodwork.
Gunners substitute Zak Ansah fired a warning shot on 65 minutes when he forced a good save from Archer after leading a break forward, but we comfortably dealt with the resulting corner.
Chiriches was getting forward at will as we continued to boss the contest and it was he who was tripped by Gedion Zelalem right on the edge of the visitors' box mid-way through the half. Lamerias, though, was unable to cap a fine performance on the night with a goal as he sent the set piece over the top.
Oduwa and then Chiriches forced full-stretch diving saves from Macey from roughly the same area on the left-hand side before Coulibaly tested the stopper once more as the game entered its final quarter of an hour.
Macey spooned another Lamerias effort behind for a corner shortly afterwards, before the same player was off target after a clever one-two with Winks.
Our second goal finally arrived six minutes from time and it was nothing more than we deserved.
Oduwa's storming run down the left led to the ball being switched to McEvoy on the opposite flank, who in turn laid it off for Winks to drive home from the edge of the box.
Coulibaly cracked a shot against the upright as we chased a third, while Chiriches was denied by the goalkeeper after a near 70-yard run as we rounded off a dominant performance still in command.
Spurs: Archer, A McQueen (McEneff 82), Ogilvie (Gallifuoco 85), Veljkovic, Ball (Stewart 77), Chiriches (c), McEvoy, Winks, Coulibaly, Lameiras, Oduwa. Subs (not used): Dombaxe, Priestley (GK).
Arsenal: Macey, Siemann, Ormonde-Ottewill, Kamara, Ajayi, Hayden (c), Iwobi (Jebb 79), Zelalem, Lipman (Ansah 61), Eisfeld, Toral. Subs (not used): Smith, Mugabo, Huddart (GK).