Lyndon Dykes’ 30th-minute penalty ensured Steve Clarke’s men held on for victory at the Ernst Happel Stadion and moved into second place in Group F.
In Tuesday night’s other results, Denmark thrashed Israel 5-0 and the Faroe Islands defeated Moldova 2-1 in Chisinau.
What do Scotland need to do to qualify?
There are four rounds of matches remaining in Group F.
Denmark are top with 18 points, seven clear of Scotland in second place. Israel are a point behind in third place, while Austria are fourth on seven points. The Faroes have four points, and bottom-team Moldova have one.
The top team qualifies automatically for the World Cup, while the second-placed team moves into the play-off stage.
The ten European group runners-up will be joined by the best two Nations League group winners, based on the Nations League overall ranking, that finished outside the top two of their qualifying group.
Then those 12 teams will be drawn into three play-off paths, playing two rounds of single-match play-offs (semi-finals and finals, with the home teams to be drawn), with the three path winners qualifying for the World Cup.
As Denmark are so far ahead, they are landslide favourites to top the group. Scotland are all but playing for second place.
What are the next fixtures?
Scotland's next match is a crucial meeting with Israel at Hampden on Saturday, October 9. With Scotland now a point clear of the Israelis, it will offer Clarke's men a real chance to put daylight between themselves and their nearest challenges before travelling to Torshavn to take on the Faroe Islands on Tuesday, October 12.
Scotland's penultimate Group F match is away to Moldova in Chisnau on Friday, November 12, before they welcome the Danes to Hampden on Monday, November 15 to complete the campaign.
Victories over Israel, the Faroes and Moldova would assure Scotland a place in the play-offs.