The result has left Luis de la Fuente's side at the top of Group 4 of League A, boasting seven points from their opening three matches, while Denmark have dropped to second, having now collected six points from three games.
Zubimendi had a half-chance for Spain in the 19th minute, but his effort struck the side-netting, before Alvaro Morata just missed the post shortly after during a period of domination from La Roja.
Denmark improved as the first period developed, with Christian Eriksen calling the shots, but the final chance of the first period also fell to Spain, with Lamine Yamal smashing over the crossbar after smart work from Pedro Porro.
Fabian Ruiz fired one into the side-netting early in the second period, and it was all Spain approaching the hour, but Denmark were riding the storm impressively, with the visitors defending their own penalty box like their lives depended on it.
Denmark goalkeeper Kasper Schmeichel made a brilliant save to deny Morata just past the hour, before the same two players collided shortly after, with the stopper again coming out on top.
However, the breakthrough finally arrived in the 79th minute of the contest, with Zubimendi volleying into the back of the net from outside the box, with Schmeichel allowing the ball through his grasp.
Spain had 25 attempts during a dominant performance, and it would have been tough to take had De la Fuente's side not collected all three points, but they got there in the end due to Zubimendi's effort.
SPAIN VS. DENMARK HIGHLIGHTS
79th min: Spain 1-0 Denmark (Martin Zubimendi)
Spain make the breakthrough in the 79th minute of the contest, and the goal comes from Zubimendi.
It is a volley from the Real Sociedad midfielder, as he sends one towards the goal from outside the box, but Denmark goalkeeper Schmeichel might feel that he should have kept it out.
MAN OF THE MATCH - MARTIN ZUBIMENDI
Zubimendi was the match-winner on Saturday evening with his volley, but it was the midfielder's overall performance that impressed the most, as he excelled in the absence of Rodri.
The 25-year-old, who turned down the chance to join Liverpool from Real Sociedad over the summer, boasted a pass success rate of 93%, while he also made two tackles during an all-action performance.
SPAIN VS. DENMARK MATCH STATS
Possession: Spain 61%-49% Denmark
Shots: Spain 25-3 Denmark
Shots on target: Spain 6-1 Denmark
Corners: Spain 4-4 Denmark
Fouls: Spain 9-17 Denmark
BEST STAT
WHAT NEXT?
Spain will now switch their attention to Tuesday's clash against Serbia, before taking on Denmark again and Switzerland in their final two matches in the section next month.
Denmark, meanwhile, will be aiming to bounce back from their first defeat in the group when they head to Switzerland on Tuesday, before tackling Spain and Serbia in their last two games in Group 4 of League A in November.