Rasmus Højlund scored the only goal of the match from the penalty spot in the 75th minute as Denmark completed a league double over Wales following their 2-0 victory in Copenhagen last week.
Wales and Denmark battle for control
Both sides struggled to create clear-cut chances during a cagey first half, although Denmark carried the greater attacking threat.
Wales had slightly more possession but Denmark looked dangerous whenever they moved forward, with Rasmus Højlund and Gustav Isaksen causing problems for the home defence.
Morten Hjulmand headed wide, while Isaksen also missed two opportunities as Denmark continued to look for an opening.
The visitors went into half-time with the match still goalless.
Højlund breaks the deadlock
Denmark eventually found the breakthrough in the 75th minute after Wales goalkeeper Danny Ward made a costly error.
Ward gifted possession to Isaksen inside the penalty area before rushing out to recover. The goalkeeper brought down the Denmark forward, leaving the referee with little choice but to award a penalty.
Højlund stepped up and calmly converted from the spot to give Denmark a 1-0 lead.
Denmark hold on for important victory
Wales attempted to respond in the closing stages, but Denmark's defensive organisation prevented the hosts from finding an equaliser.
The result marked Denmark's second consecutive victory over Wales in the space of a week and further strengthened their position in the tightly contested Group A4.
Denmark are now second in the group on nine points, three behind leaders Portugal, while Wales remain on three points.
Wales 0-1 Denmark
• Wales: No goals
• Denmark: Rasmus Højlund 75' (pen)
The victory leaves Denmark in a strong position heading into the final two Nations League matches in November, while Wales face a tougher task after suffering their third defeat in four group matches.