Milivoje Novakovic converted a penalty with 11 minutes remaining but Handanovic proved the difference as Slovenia held on for a 1-0 victory in Group E action in Maribor on Thursday.
Handanovic thwarted the visitors on numerous occasions, keeping out Granit Xhaka's powerful volley early on.
The towering Inter keeper was also on hand to deny Haris Seferovic, Ricardo Rodriguez and Ricardo Rodriguez while he dealt with countless crosses as Switzerland went in search of an equaliser during the closing stages.
Katanec reserved special praise for the 30-year-old as Slovenia ended a four-game losing streak.
"Of course we are happy with these three points," Katanec said afterwards. "Congratulations to the boys.
"We knew that Switzerland are a very strong team, but we had a very high-level goalkeeper tonight. He saved us when it was most difficult."
Katanec paired Novakovic and Zlatan Ljubijankic up front in the first half, a decision the coach regretted after watching Switzerland dictate proceedings.
But Ljubijankic made way for midfielder Jasmin Kurtic as Slovenia bolstered their midfield stocks and it proved to be a masterstroke.
"I started with two forwards, as you saw in the first half – that turned out to be a wrong decision," said the 51-year-old.
"With all three substitutions we caused problems for Switzerland. But of course we can play better than this.
"We have to be physically better in our next matches. Now we go to Lithuania for another three points."
Meanwhile, Kevin Kampl urged his team-mates to build on the result after battling to victory.
"We have to be very happy with these three points," said Kampl.
"We've shown a fighting spirit and the crowd were behind us. We have to win now in Lithuania [on Sunday] or this result will be meaningless."
