Ikechi Anya's breakaway goal after 10 minutes, reminiscent of his strike in Dortmund against Germany, proved to be the decisive moment in Prague.
Scotland might have had a second had Alan Hutton been awarded what looked like a clear-cut second-half penalty.
But Gordon Strachan's preparations for the World Cup qualifiers began with only a second-ever win over the Czechs.
They return home to face Denmark in another friendly at Hampden Park on Tuesday, with further matches in June against Italy and France before they begin World Cup qualifying against Malta in September.
Scotland manager Strachan spoke before the match of having picked difficult friendly opposition to help his players develop ahead of more meaningful fixtures.
The Czechs certainly provided a stern test, reinforcing their status as winners of a tough group in the Euro qualifiers.
They bombarded Allan McGregor's goal in the opening 20 minutes, only for the Hull goalkeeper to produce a string of saves to deny them.
He palmed Tomas Sivok's header round the post after Borek Dockal had smashed a free-kick off the bar.
A diving save to tip away Martin Frydek's rasping shot was the other highlight after Anya had bemused the home supporters by slotting Scotland in front.
The Watford man raced onto Ross McCormack's through ball down the left flank and finished strongly, first time, with his left foot.
The Scots' first-half display was littered with slack passes and a lack of concentration.
But they began to get the Czechs' measure and became more composed as the match progressed.
A flurry of substitutions on both sides also had a disruptive effect but Scotland looked assured in the second period and might have gone further ahead when substitute Matt Phillips saw his header from a Robert Snodgrass cross saved by Tomas Koubek.
But McGregor was called upon once more to make another top-class intervention, turning an effort by Lukas Marecek round the post.
Although the Scotland management team had stressed this was not an exercise in experimentation, Strachan took the opportunity to give Aberdeen midfielder Kenny McLean a debut.
Playing just behind McCormack, McLean found possession hard to come by in what was a tough introduction to international football, which lasted 57 minutes.
McCormack, given a rare chance in his first cap for two years, also lacked service but played a crucial role in Anya's goal, showing great strength and weight of pass to play the winger through.
There were late debuts too for striker Tony Watt and defender Paul Caddis, which would have told the manager little about their ability to handle this level of football, but offered them a moment to savour.
Match ends, Czech Republic 0, Scotland 1.
Second Half ends, Czech Republic 0, Scotland 1.
Corner,  Scotland. Conceded by Pavel Kaderábek.
Attempt missed. Matej Vydra (Czech Republic) right footed shot from the right side of the box is high and wide to the right. Assisted by Lukás Marecek.
Foul by Daniel Pudil (Czech Republic).
Paul Caddis (Scotland) wins a free kick on the right wing.
Substitution, Scotland. Paul Caddis replaces Ikechi Anya.
Substitution, Czech Republic. Jakub Rada replaces Vladimir Darida.
Foul by Matej Vydra (Czech Republic).
Russell Martin (Scotland) wins a free kick in the defensive half.
Attempt missed. Matthew Phillips (Scotland) header from the centre of the box misses to the right. Assisted by Ikechi Anya with a cross.
Foul by Daniel Pudil (Czech Republic).
Tony Watt (Scotland) wins a free kick on the right wing.
Attempt missed. Lukás Marecek (Czech Republic) left footed shot from outside the box is too high following a corner.
Corner,  Czech Republic. Conceded by Allan McGregor.
Attempt saved. Daniel Kolár (Czech Republic) right footed shot from the centre of the box is saved in the bottom right corner. Assisted by Vladimir Darida with a cross.
Attempt missed. Matthew Phillips (Scotland) right footed shot from outside the box is high and wide to the right.
Offside, Czech Republic. Pavel Kaderábek tries a through ball, but Matej Vydra is caught offside.
Substitution, Czech Republic. Daniel Pudil replaces Josef Sural.
Substitution, Czech Republic. Lukás Marecek replaces Kamil Vacek.
Substitution, Scotland. Tony Watt replaces Ross McCormack.
Josef Sural (Czech Republic) wins a free kick in the defensive half.
Foul by Matthew Phillips (Scotland).
Foul by Kamil Vacek (Czech Republic).
Robert Snodgrass (Scotland) wins a free kick in the defensive half.
Tomas Sivok (Czech Republic) wins a free kick in the defensive half.
Foul by Robert Snodgrass (Scotland).
Jiri Skalak (Czech Republic) is shown the yellow card for a bad foul.
Foul by Jiri Skalak (Czech Republic).
Charlie Mulgrew (Scotland) wins a free kick on the left wing.
Foul by Daniel Kolár (Czech Republic).
Christophe Berra (Scotland) wins a free kick in the defensive half.
David Limbersky (Czech Republic) wins a free kick in the defensive half.
Foul by Robert Snodgrass (Scotland).
Attempt missed. David Limbersky (Czech Republic) header from the centre of the box is high and wide to the right. Assisted by Kamil Vacek with a cross.
Attempt missed. Robert Snodgrass (Scotland) header from the left side of the box misses to the right. Assisted by Matthew Phillips with a cross.
Foul by Daniel Kolár (Czech Republic).
Robert Snodgrass (Scotland) wins a free kick in the defensive half.
Attempt missed. Matej Vydra (Czech Republic) right footed shot from outside the box is close, but misses to the left. Assisted by Josef Sural.
Substitution, Czech Republic. Matej Vydra replaces Tomas Necid.
Scotland withstood heavy early pressure to record an impressive friendly win over Euro 2016-bound Czech Republic.