Sock, of America, took just over two hours to see off his French opponent 7-5 7-6(5) and reach the last four in the Swedish capital.
It was a shaky start for Sock, but he saved three break points in the opening game before holding to move ahead.
Sock again held firm under pressure from Simon later in the set and then snatched it 7-5 by converting his first break point.
At the start of the second set, Simon broke to love to get back on track and at 5-2 it seemed that he would take the match to a decider.
However, Sock won four games in a row, saving a set point along the way, and this time a tie-break was needed.
At 3-3 it was evenly poised as the duo changed ends, but Sock then won three straight points to earn three match points and he converted the third to reach the semis, where he will face either Jeremy Chardy or Richard Gasquet, both of France.