It was the first match of Moller's career on grass at all, and with the lack of experience, it wasn´t a surprise that the world number 12, Frances Tiafoe, was too big a challenge for the young Dane.
Tiafoe was on top throughout the match, and won comfortably without showing his best tennis, 6-3, 6-4, 6-2.
Moller returned better than he served, and briefly gave examples of his feared backhand, but it wasn't enough to challenge Tiafoe.
The Dane's big problem was the serve as he only managed to get 29% of his first serve in play during the first set and that was't nearly enough against Tiafoe.
Moller briefly had the feeling of settling into the match, when he broke Tiafoe's serve for 2-3 but the strong American immediately responded and had no problems closing out the set 6-3.
The Dane improved his serve in the second set but Tiafoe was still able to break Moller's serve in the beginning of the set and as the American did not give much away in his own service games, Tiafoe won 6-4 without any major problems.
There was almost a full house at Court No 12, which can accommodate a little more than 1,700 spectators. Many of them tried to back the Dane hoping to get a more exciting match, but the drama was completely absent.
It was symbolic of Moller's performance that it was a double fault, which paved the way for Tiafoe breaking the Dane's serve for 2-1 in the third set.
And from then on the Dane fell apart. Moller's eighth double fault helped Tiafoe to a 4-1 lead, and soon after the American converted his first match point to book his ticket for the next round
Moller can take comfort from the fact that he has played in two successive Grand Slam tournaments as he look to break into the Top 100.