Despite some incredibly close games in the second set, Draper was beaten 6-3 7-5 by world number 21 Frances Tiafoe in an hour and 24 minutes.
Tiafoe started well in the opening set, but Draper hit two aces in the fourth game to draw level at 2-2.
Although the American was threatening to pull away, Draper put up a fight in the eighth game, which lasted over 10 minutes, before Tiafoe eventually broke and went on to take the first set.
World number 45 Draper began the second set well, firing down four aces across the first three games, but Tiafoe kept pace with his opponent.
With the scores level at 5-5, the match looked to be heading towards a tie-break, but Tiafoe got the edge on Draper to break in the 11th game before going on to win the second set 7-5.
Evans was also knocked out of the tournament as he fell to a 6-3 6-4 defeat to fifth seed Stefanos Tsitsipas.
The world number five displayed his dominance, smashing three aces in the third game to go 3-0 up, before Evans rallied to bring the score to 5-3.
The Greek showed his strength, however, capitalising on a double fault from the the world number 27 in the ninth game to take the first set.
Tsitsipas broke in the first game of the second set, but faced a difficult challenge against Evans, who came back to trail 5-4.
However, Tsitsipas was able to hold on and saw the match out with an ace in the 10th game to progress to round three, where he will face either Corentin Moutet or British world number 13 Cameron Norrie.