Southampton controlled large parts of the game at the Emirates Stadium as they handed the Gunners their first defeat in all competitions since the first weekend of the season, a run stretching 19 fixtures.
Arsene Wenger made 10 changes to the side which started the 3-1 victory over Bournemouth at the weekend and the upheaval proved costly during a first half where the Gunners failed to register a single shot on goal.
Southampton were not firing on all cylinders but they were a constant threat on the counter-attack through Shane Long and Sofiane Boufal, and the pressure they were putting the Arsenal defence under paid off in the 13th minute.
After a shot in the penalty area was blocked, the ball fell to Clasie who fired into the top corner from 16 yards out to net his first goal for the Saints.
Arsenal were still having plenty of possession but the likes of Lucas Perez, Alex Iwobi and Aaron Ramsey were creating nothing in the final third, and it came as no surprise when Southampton doubled their lead with seven minutes of the first half remaining.
It was club-record signing Boufal who provided the assist, with his pass into the penalty area being collected by Bertrand, who took a touch before finding the bottom corner from 14 yards.
Soon after the restart, Arsenal finally registered their first effort on goal but Perez's left-footed effort inside the penalty area lacked power and Fraser Forster claimed easily at the near post.
Ramsey then failed to make the most of an opportunity from 12 yards out before Southampton went back on the offensive with Long forcing a sprawling save out of Emiliano Martinez.
Moments later, Boufal came within inches of sealing victory for the Saints when his low drive narrowly missed the far post, but Southampton were not left ruing that opening as Arsenal struggled to test Forster.
Southampton could have further extended their lead through Long and Steven Davis late on before Alex Oxlade-Chamberlain fired harmlessly over for Arsenal with 11 minutes left when he probably should have scored.
The visitors finished strongly with Pierre-Emile Hojbjerg forcing a save out of Martinez but they were more than content to settle for the two as they reached the last four of the competition for the first time since 1987.