The visitors, who have one eye on the 2018 World Cup qualifiers, were under large spells of pressure during the game, but Anya's 10th-minute strike was enough to claim back-to-back away wins for the first time since March 2014.
Borek Dockal stepped up to take it and his effort came close to rippling the goal, but he just missed the target as the ball clattered off the crossbar.
Two minutes later, Scotland were celebrating Anya giving the team the lead as he received a through-ball from Ross McCormack on the counter-attack and slotted it under the goalkeeper.
Despite taking the lead, Gordon Strachan's side were under intense pressure from the Czechs as the hosts attempted to find an equaliser.
The visitors had a handful of chances after weathering the storm, and debutant Kenny McLean came close at the back post after trying to latch on to McCormack's cross, but the ball was headed away by Tomas Sivok before the midfielder could get a touch.
Scotland had a decent five-minute spell as the first half drew to a close, with Robert Snodgrass delivering some teasing crosses into the box, but none of the chances were converted.
After the break, the hosts came out with attacking intent and immediately put their opponents under pressure as Pavel Kaderabek found space in the box before drilling a low shot, but it was deflected on the way and McGregor claimed with ease.
Scotland were forced to defend their box in quick succession as David Limbersky delivered a dangerous cross towards Tomas Necid, who was put off by Berra's presence while trying to connect with a header.
Alan Hutton, who was making his 50th international appearance, marshalled the defence well, and at the other end, McLean almost had a dream debut when he shot low from a tight angle, but his effort just trickled wide of the post.
Scotland began to pose a threat of their own and goalkeeper Tomas Koubek was called on to prevent a second goal when Snodgrass found Matt Phillips, who headed towards the net, but the keeper was able to grab the ball in the air.
The visitors felt that they were unlucky to be denied a penalty after Hutton was wiped out by Limbersky in the box, and the same fate was dealt to the home side after Matej Vydra went to the ground after a challenge from Berra.
Despite chances at both ends, the game ended in a slender win for the Scots.