Needing a draw to qualify, the Swiss went ahead after 10 minutes through Itten, who had scored on his debut on Friday to give them a 1-0 win over Georgia. Ruben Vargas got to the byline and sent over a low cross which the Gibraltar defence failed to intercept and Itten tapped in.
Vargas himself headed the second from Loris Benito's cross five minutes after halftime and Christian Fassnacht added the third seven minutes later after the Gibraltar defence failed to deal with Michael Lang's cross.
Gibraltar pulled one back in the 74th minute when a long throw was headed on at the near post, the ball bobbled around and Reece Styche prodded it in.
But the celebrations did not last long as Switzerland went straight down the other end from the re-start and Benito added a fourth.
Itten took his tally to three in two internationals when he headed the fifth and provided the pass for Granit Xhaka to slot home the sixth as Gibraltar tired.