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Ikechi Anya scored with Scotland's first shot of the game.
Scotland have scored in 10 of their last 11 away internationals.
Gordon Strachan has overseen successive away wins as manager for the first time since March 2014 (a run of three wins).
Scotland have also kept consecutive clean sheets for the first time since March 2014.
The Scots have lost just two friendlies under Strachan (W6 D2).
So it's on to Hampden on Tuesday where Denmark will be the visitors. What a fillip it would be if Scotland could register another win, albeit in a friendly.
Gordon Strachan: "The players can be pleased with themselves, I don't think anybody needs to be worried about their performance."
Gordon Strachan: "After a low start in the first half I thought the Czech Republic played very well but I think we allowed them to play. I thought McGregor was fantastic. We didn't close people down. Once we sorted that out in the second half and passed the ball it became a different game. The group have made it a good night. We are not a great side but we can turn ourselves into a decent side by working hard. It was great learning in the first half about how small things can change a game."
Scotland manager Gordon Strachan: "The win." There you have it.
Scotland's Under-21s are trailing 2-0 against France, a damaging blow to their hopes of qualifying for the European Championships.
BBC Scotland's Tom English: "That's a strong victory for them against a good side. The Czech Republic ran out of ideas in the second half. They limited the Czech Republic to the odd chance. McGregor was in top form and Scotland should have had a penalty. It's a very creditable victory and it's a good confidence booster."
- Quote Message: It is more than a decent result, it's a fantastic result. Gordon Strachan has had a bit of criticism for his squads. Everybody has been fantastic. from Former Scotland striker Darren Jackson
Tony Watt shows good strength and an old head to kill time down in the left corner. What's more, he wins a corner. Terrific.
The Tartan Army are in good voice, with a hearty rendition of Flower of Scotland. We're in to the three minutes of injury time now.
Birmingham right-back Paul Caddis comes on for Ikechi Anya. He joins Kenny McLean and Tony Watt in making his Scotland debut.
Barry Bannan loops a cross over from deep on the left and Ikechi Anya heads wide, under pressure at the back post.
Now here's a story! Veteran Athletic Bilbao striker Aritz Aduriz has bagged on his first start for Spain - six years after his only other international cap.
The 35-year-old frontman, who has netted 31 goals for the La Liga side in all competitions, slams in an equaliser just seconds after Italy take the lead through Napoli's Lorenzo Insigne.Copyright: Getty Images
All over in Odessa. So what, I hear you cry? So Ukraine - Wales' next opponents and Northern Ireland's Euro 2016 rivals - have earned a steady if unspectacular 1-0 win against Cyprus. Shakhtar Donetsk midfielder Taras Stepanenko bagged the first-half winner.