After a goalless first half, the Danes finally made the breakthrough when Joakim Mahle fired home from close range after being set up by Thomas Delaney on 53 minutes.
The win moves Denmark seven points clear of second-placed Scotland with just two matches remaining, meaning their spot in Qatar is securing.
Denmark have a 100 per cent record in Group F, scoring 27 goals and not conceding any so far, while Austria now cannot make the top two of the section.
In the pool’s other match, Israel – stung after losing to Scotland on Saturday in heartbreaking circumstances – knew that only a win would do against Moldova in Be’er Sheva. The visitors went into the match with only one point to their name, offering Willi Ruttensteiner’s men the perfect opportunity to bounce back.
The hosts had to keep their patience, but were rewarded on 28 minutes when, after good work from Manor Solomon, key attacker Eran Zahavi scored to give the Israelis a 1-0 lead.
Zahavi then turned provider four minutes into the second half, teeing up Munas Dabbur to make it 2-0.
Moldova finished the night with ten men after Oleg Reabciuk was dismissed for a second bookable offence, although they grabbed a late consolation via Ion Nicolaescu.
Israel are the only team that can move Scotland from second spot, but they trail them by four points.
In the next round of fixtures – when Scotland travel to Chisinau to face Moldova – Austria host Israel in Vienna, while Denmark welcome the Faroe Islands to Copenhagen.