The visitors came out of the traps quickly, and came close after only three minutes when a ball across the face of goal from Jonas Wind just evaded Rasmus Hojlund.
Having netted a hat-trick against Finland on Thursday, Hojlund then came close to setting up a goal when his backheel played Joakim Maehle in, but his shot was saved before rebounding off the striker and going wide.
The Danes didn’t have to wait long to take the lead though, as a pinpoint ball over the top from captain Simon Kjaer played Hojlund in, and the youngster delicately lobbed the ball over Igor Shatskiy to make it 1-0.
The hosts had a good chance to equalise a couple of minutes later but defender Serhiy Malyi’s header was straight at Kasper Schmeichel, and the hosts were made to pay 10 minutes before the break when Denmark worked their way into the area and saw a shot rebound to Hojlund, who poked the ball home from close range for his fifth goal in two games.
Kazakhstan may be the second-lowest ranked nation in Group H, but they remained very much in the contest and caused Denmark problems in the second half.
Abzal Beysebekov had a shot from inside the area pushed onto the right post, and then with just under 20 minutes remaining the Hawks were awarded a penalty when referee Novak Simovic adjudged that the ball had hit the arm of Jonas Wind from a corner.
Bakhtiyar Zaynutdinov made no mistake to set up a grandstand finish in Astana.
All of the momentum was with Kazakhstan in the closing stages, and in dramatic circumstances they turned the game on its head as a thunderbolt from Askhat Tagybergen found the top right corner, before substitute Aymbetov headed home to send the Astana crowd into raptures and secure a famous win which blows Group H wide open.
The drama wasn't over there for Aymbetov though, as he was booked twice in stoppage time and given his marching orders before the final whistle sounded on a classic.
Flashscore Man of the Match: Abat Aymbetov (Kazakhstan)
