It took Giron just one hour and 21 minutes to claim a 6-3 6-4 victory over the world number 13 and all but end Norrie's slim chances of reaching next month's ATP Finals.
Giron, who sits 45 places below Norrie in the rankings, now progresses to the quarter-finals of the competition where he will meet either Andrey Rublev or Grigor Dimitrov.
After a tight start to the opening set, Giron pulled away with his second break of the match to take the eighth game and went on to win the first set 6-3.
The second set was again closely contested until, with the score tied at 3-3, Norrie double-faulted, allowing Giron to break and take the seventh game.
The American looked likely to pull away again with a lead of 5-3, but Norrie served back-to-back aces to win the ninth.
Despite the British player's best efforts, Giron came from 15-40 down in the 10th game to win the match and reach the last eight.