Youzhny, runner-up to Roger Federer in the ATP 250 grass-court event 12 months ago, was dumped out by big-serving Ivo Karlovic 6-3 6-4.
The Russian, seeded sixth for the tournament in Halle, was broken in his very first service game as Karlovic raced into a 4-1 lead.
From there, the Croat had little trouble in seeing out the set, holding to love in game nine to move in front in the match.
The second set followed a similar pattern, Youzhny once again broken at the first time of asking, and the Russian won just one point on the Karlovic serve throughout the remainder of the set.
Karlovic will now take on Lu Yen-hsun in the second round, after the world number 48 saw off Mate Pavic 7-6 (7-2) 6-7 (11-13) 7-6 (7-5) in a closely fought encounter.
Gasquet was another high-profile casualty, going down 6-4 6-4 to Robin Haase of the Netherlands.
The Dutchman made the crucial breaks in game seven of the first set and game three of the second, with fifth seed Gasquet unable to force as much as a break point against his opponent.
Haase will now take on Alejandro Falla, who overcame Lukasz Kubot 6-2 3-6 6-3 to progress to the second round.
Elsewhere, Gael Monfils booked a meeting with fourth seed Kei Nishikori with a routine 6-1 7-5 victory over Benjamin Becker.
Following the withdrawal of Tommy Haas with a shoulder injury, Albano Olivetti was handed a lucky loser's spot, but he failed to make the most of his opportunity, losing out 6-4 6-2 to Steve Johnson.
Johnson will meet Teymuraz Gabashvili in the next round, after the Russian came through 6-7 (9-11) 6-4 7-6 (7-4) against Illya Marchenko.