The Hoops had the best of the chances but their visitors managed to frustrate them for large spells of the game, with goalkeeper Jesper Christiansen in fine form.

In the end, a spot kick 10 minutes before the break, which Miller despatched clinically, proved to be all that separated the two teams.

An horrendous defeat to Artmedia Bratislava in the Champions League qualifiers robbed Celtic of European football last year and it was clear they were going to relish every second of their involvement in the Champions League this season.

The Scottish champions opened the group stage campaign with a brave performance against Manchester United but the 3-2 reverse at Old Trafford meant they were well aware of how important a win was tonight.

Celtic may have aspirations of following in the footsteps of rivals Rangers by progressing from the group stages to the knockout phase.

But there was no chance of them underestimating the threat of a Copenhagen side who had already disposed of Ajax in the qualifiers, before taking a point from their opener against Benfica.

The Danes already knew they would be without Jesper Gronkjaer, after he suffered a groin injury in that match in Copenhagen.

And Celtic were also deprived of one of their key players when the ankle injury picked up during the Old Firm triumph on Saturday kept Jan Vennegoor of Hesselink out of this match.

In came Maciej Zurawski in place of the towering Dutch striker but it was Miller who showed signs of stealing the show after finally breaking his Celtic duck against Rangers.

He latched on to a great ball from Paul Telfer and raced into the box, only for his shot to cannon off the body of Michael Gravgaard, before Shunsuke Nakamura was also denied by the defender.

Miller then connected with an Aiden McGeady cross but sent his header flashing wide of the target.

Celtic had the ball in the back of the net when the striker turned provider to set up fellow hitman Zurawski but the offside flag was already raised before the shot had even left his boot.

McGeady then tried his luck with a rasping low effort, which fizzed past the upright and crashed into the advertising hoardings.

Copenhagen had seemed content to sit back and soak up the pressure but the breakthrough finally came after 35 minutes when Nakamura was tripped in the box by Gravgaard.

Miller – clearly brimming with confidence after ending his goal drought - stepped up for the pressure kick and coolly sent the ball low past goalkeeper Christiansen.

Christiansen’s brief spell at Rangers meant he was booed every time he touched the ball and the Celtic support had even more reason to be unhappy with the Dane when he pulled a save out of the top drawer to deny Miller’s header before the break.

With only one goal in it at half-time, neither side opted to make any changes at the break.

McGeady was causing plenty of problems for Copenhagen with his pace after the restart and one challenge, from Gravgaard, prompted an angry reaction from the Celtic bench.

Celtic claimed another penalty with an hour gone when Stephen McManus appeared to be pushed in the box by Brede Hangeland but this time the pleas were ignored.

Then it was Copenhagen’s turn to put Celtic under pressure when Artur Boruc was called into action to prevent Hjalte Norregaard’s shot finding the back of the net.

But, again, the plaudits went to his opposite number Christiansen.

McGeady slipped the ball through to Miller, who should have netted, but somehow found himself denied by the knees of the goalkeeper.

Copenhagen pushed forward in numbers in search of an equaliser as the game raced towards its conclusion.

And Atiba Hutchinson was not too far away with an effort which trickled just past the post.

But it was too little too late from the Danish champions as Celtic got off the mark with three points in Group F.