Swiss skipper Alexander Frei was twice denied by Petr Cech before he was forced off injured and Hakan Yakin headed a chance wide after the break.
On 70 minutes Sverkos slotted in a low volley from 12 yards after latching on to a ball over the top of the defence.
The Swiss were denied an equaliser when Johan Vonlanthen's volley hit the bar. And it might prove that the width of the woodwork could be the end of the Swiss, with tough matches against Portugal and Turkey to come in Group A.
Realistically the tournament's co-hosts needed to get something out of their opening game, and they started as though they were absolutely determined to.
After three minutes, the Czechs gave the ball away cheaply in midfield but Frei dragged his shot wide and failed to capitalise.
That should have been a wake-up call to the Czechs, but they appeared in no rush to raise their tempo.
As a result, Switzerland - unfancied in the group - grew in confidence and stroked the ball around, but with little cutting edge.
A weak effort from Gokhan Inler was indicative of their toothless attacks up until that point.
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Frei, though, looked the liveliest player on the pitch, and despite having no help from his struggling strike partner Marco Streller, he began to cause some problems.
It took a decent stop from Cech to deny him the opening goal of Euro 2008 after he had latched on to a long clearance from his own keeper.
Frei followed that up with a 30-yard shot that Cech opted to use his fists to clear, despite there being little sign of the any movement from ball.
Sadly for Frei it would prove to be virtually his last contribution to the match as he later hobbled off after a jarring challenge.
The Czech Republic had already started showing signs of waking from the slumber before then, and with crisp passing and better movement were dictating the play.
But their final ball was awful and up front, giant striker Jan Koller looked slow and cumbersome.
After the break the Swiss brought on Hakan Yakin for the injured Frei and they started the half positively.
As the Czechs found themselves pinned back, Yakin got himself into the action but was too high with an effort following a free-kick.
But it was clear Yakin was already having more of an impact than Koller and it was no surprise when the Czech's top scorer was replaced by Vaclav Sverkos.
And the Swiss substitute should have opened the scoring when he was presented with a free header, but he directed the ball wide the right-hand post.
It was to prove a costly miss when, on 70 minutes, Czech substitute Sverkos finished smartly into the bottom corner after latching on to a ball over the top of the Swiss defence.
Yet Switzerland almost snatched a draw when Tranquillo Barnetta's shot was superbly parried by Cech and Vonlanthen smashed the follow-up against the woodwork.