Although the home side were not necessarily at their best on Saturday afternoon, an early goal from John McGinn and two late efforts from Manchester United midfielder Scott McTominay sealed a deserved win over opponents who had Nicholas Ioannou sent off for two yellow cards in the final seconds.
A day after it was announced that Steve Clarke had penned a new long-term contract, his side were under pressure to deliver a result that felt imperative if they are to challenge group rivals Spain and Norway.
After a positive start, Aaron Hickey was denied at the near post, but it was not long until the Tartan Army were left to celebrate an opening goal.
Andrew Robertson was the provider, his deflected cross from the left finding McGinn at the back post, and the Aston Villa man had the simple task of converting from close range.
Cyprus were offering a threat at the other end, however, and captain Kostakis Artymatas sent a dipping shot over the crossbar from distance.
Che Adams had an opportunity to double Scotland's lead before the break, only for the forward's header to be tipped over by Demetris Dimitriou.
While Scotland threatened a second goal after the break, they were failing to find the final pass or have shots blocked by strong defending from the visitors.
Debutant goalkeeper Angus Gunn retained his clean sheet at the other end, keeping out a deflected effort from Andronikos Kakoullis, and it proved to be crucial as Scotland eventually sealed maximum points with three minutes left.
Three substitutes were involved as Ryan Christie and Lyndon Dykes combined to tee up McTominay, who smartly finished past Dimitriou having latched onto Dykes's knockdown.
Just when it seemed like the game would peter out, McTominay notched his second, the Man United midfielder side-footing home after being teed up by Robertson to cap an enjoyable afternoon ahead of a sterner test with Spain.