It is the first victory that the Irish nation have claimed over Switzerland since 1992, when they defeated a team coached by now-England manager Roy Hodgson.
The ball was whipped in towards Shane Duffy, who kept the move alive by heading towards goal and finding Clark, who nodded home from close range.
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After Ireland took the lead, Switzerland ramped up their attack and had a wave of chances, one of which was created by Blerim Dzemaili, who unleashed a shot from the edge of the area with his first touch, but it was too high and surged over the crossbar.
Haris Seferovic showed some individual skill by turning inside the box before firing towards the net, but his effort just trickled wide of the post.
The Swiss kept knocking on the door and both Dzemaili and Seferovic were involved again as the midfielder whipped in a corner and found his teammate, who headed over the bar.
The hosts were dealt a blow when Kevin Doyle was stretchered off just before the half-hour mark following a heavy challenge from Timm Klose, but they managed to keep Switzerland at bay for the remainder of the first half.
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O'Neill's side had a few opportunities to double their lead before the half-time whistle was blown, the closest of which came from Shane Long, who latched on to Seamus Coleman's cross but his header was denied by the crossbar.
After the break, the game mirrored the first half as Switzerland had all the possession but failed to get the ball past goalkeeper Darren Randolph.
The hosts stayed tight and solid at the back, preventing their opponents from finding a way through, and Clark did his bit to hold off the Swiss.
Breel Embolo managed to get into the box and was poised to shoot, but Clark timed his tackle well and made a crucial block.
Daryl Murphy had a decent attempt in the 76th minute when he charged into the box, but instead of playing in his teammates at the far post, he went for glory, and his shot bended high and wide.
Switzerland continued searching for an equaliser, but Ireland managed to hold on to their lead.