Denmark struggled to find the net after thrashing Republic of Ireland 5-1 in the qualifying play-offs and looked set for a third consecutive goalless draw in Brondby.
But after toiling in front of goal, Poulsen bent a glorious 71st-minute effort into the top-left corner to lift the mood, before Christian Eriksen netted a quickfire second.
Denmark started brightly and Nicolai Jorgensen glanced an early header wide before seeing a close-range effort blocked by Guillermo Ochoa, who soon denied new Borussia Dortmund signing Thomas Delaney.
However, Mexico, led by the dazzling Jesus Manuel Corona on the left, built some momentum and tested Kasper Schmeichel through Hector Herrera's strike.
Schmeichel saved again when Oribe Peralta volleyed on target, but Denmark regained a measure of control with the impactful half-time introductions of Martin Braithwaite and Kasper Dolberg.
Braithwaite centred for his fellow substitute to stab wide, before twice failing to beat Ochoa from a tight angle on the left.
It looked like Denmark would be frustrated once again until Poulsen cut inside from the left and lifted a gorgeous curler over Ochoa and into the net.
That was the lift the home side needed, with Eriksen bursting through on the right three minutes later to hammer a low finish past the goalkeeper and seal a fine win.