The hosts comfortably defended the lead they took after two minutes when Clark headed home his second international goal after Shane Duffy had nodded on a corner from the left.
Ireland striker Kevin Doyle, who plays for the Colorado Rapids, then suffered a nasty injury in a block tackle after 22 minutes and had to go off on a stretcher.
Manager Martin O'Neill told reporters later that Doyle had a deep cut in his leg but nothing was broken.
Shane Long almost scored another goal in the 37th minute but his header hit the bar and Swiss keeper Yann Sommer caught his follow up.
Both teams made several substitutions in a second half in which there were few chances and little excitement.
"It was important to keep a clean sheet and to get the goal was brilliant for me," said Clark. "We had worked on corners like that.
"It was an easy finish. There is a lot of competition in the squad, everyone is fighting for their place."
O'Neill said: "I know it was a friendly but it is nice to get the win because it keeps the momentum going."
Ireland, unbeaten in 10 home games under O'Neill, continue their build-up for the European Championship in France in June with a friendly at home to fellow finalists Slovakia on Tuesday.
The Irish face Italy, Belgium and Sweden at the Euros while the Swiss take on hosts France, Albania and Romania.
Switzerland coach Vladimir Petkovic left Leicester City's Gokhan Inler out of the squad for the first time in 10 years with the experienced midfielder struggling for form.
The Swiss host Bosnia in their next friendly on Tuesday.
Ireland last beat Friday's opponents exactly 24 years ago when John Aldridge's 88th-minute penalty handed current England manager Roy Hodgson his first defeat as Switzerland boss.
(Writing by Ken Ferris; Editing by Tony Jimenez)
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