Boasting a slender advantage from the first leg, West Ham would have been content with the opening 45 minutes, as Alkmaar struggled to create clearcut chances, while Lucas Paqueta rattled the woodwork for the visitors.
The Hammers were put under greater pressure in the second period, and Alphonse Areola was required to deal with efforts from Sven Mijnans and Pantelis Chatzidiakos.
However, as Alkmaar threw caution to the wind, Pablo Fornals broke away to drive the ball past Mathew Ryan to wrap up a 3-1 aggregate win and book West Ham's place in a major European final for the third time in their history.
After coming from behind to win 2-1 in the first leg, West Ham travelled to the Netherlands with a narrow lead to protect.
It was clear from the outset that the Hammers were content to soak up the Alkmaar pressure and look to strike on the counter-attack.
For all of Alkmaar's dominance in terms of possession, the English side carried the greater attacking threat in the first period.
Jarrod Bowen dragged an effort harmlessly wide from a promising position after the Dutch side failed to clear the lines.
As the contest moved past the midway point of the first period, Paqueta cut inside onto his left foot before drilling the ball against the outside of the post.
With the vociferous home support spurring them on, Alkmaar made a promising start to the second half, with a low strike from Mijnans causing Areola to make a routine save.
Declan Rice attempted to halt the home side's building momentum when he tried his luck from a distance, forcing Ryan into a smart stop.
Alkmaar continued to control most of the ball, and they tested Areola with a long-range strike from Chatzidiakos, but they began to run out of ideas against a stubborn and resolute West Ham defence.
Desperation crept in for the hosts as time run out, allowing West Ham to break away through Fornals, who fired home a 94th-minute winner to spark wild scenes of jubilation among the travelling contingent.
West Ham fans can now look forward to a trip to Prague on June 7 when they will face either Fiorentina or Basel in the Europa Conference League finals.