Denmark had to win to secure qualification in second place behind France, and were dominant in possession. They managed to turn that into chances and forced Mat Ryan into several impressive stops in the first-half, but were unable to strike past the 'keeper and advance into Group D's second place.
Elsewhere, Tunisia took a surprise lead over France in the 59th-minute, a result which would have eliminated Denmark from the tournament in itself, but Australia bagged a lead of their own shortly after when Mathew Leckie struck in on the hour-mark.
The goal sparked greater confidence among the Socceroos, who began to control the game much more than they had in the first 45 minutes.
Denmark continued to enjoy more of the ball but couldn't find a way through, and with Tunisia holding their lead while Andersen's side needed at least two goals, qualification became a sizeable challenge.
The final whistle confirmed 1-0 wins for both supposed underdogs, with Australia and France progressing from Group D at Denmark's expense.
Earlier in the tournament the Danes came back from 1-0 down against France when Andersen found Andreas Christensen in the 68th-minute, but Kylian Mbappé's brace consigned them to defeat with just four minutes of time left.
And on their first outing Andersen made his World Cup debut on the right of a back three alongside AC Milan's Simon Kjær and Barcelona's Andreas Christensen against Tunisia.
Joachim played in every minute of his country's involvement, making his World Cup debut, earning a clean sheet and an assist in the process.
Jordan Ayew remains in the tournament, as Ghana face Uruguay on Friday 2nd at 15:00. Read about our Ghanaian fan group here.