However, Motherwell defender Steven Saunders, who won a full Scotland cap against the Faroe Islands last year, was carried off on a stretcher with an Achilles injury before being taken away from St Mirren Park in an ambulance.
Scotland recovered from the early blow to record a comfortable triumph as they warmed up for their European Championship qualifying campaign, which begins next month.
Hull midfielder Cairney coolly finished off a flowing move in 20th minute and Jack and Armstrong finished brilliantly in the final quarter.
Manager Billy Stark will have been heartened by the display ahead of the qualifiers, which begin on September 5 against Bulgaria at St Mirren Park, their home for the whole campaign.
But he will almost definitely have to do without Saunders, whose injury will be a serious worry to Motherwell manager Stuart McCall, who has been trying to add defensive cover to his thin squad.
The game was less than four minutes old when the 20-year-old collapsed as he waited for a pass with nobody near him and went down clutching his left ankle.
The physio immediately signalled for the stretcher after running on to examine the damage and Rangers defender Ross Perry came on his place.
Scotland struggled to find their rhythm with the visitors looking the more dangerous side but Stark's boys improved after a flowing 16th-minute move.
Johnny Russell took a pass from Jordan Rhodes in his stride and burst forward before unleashing a left-footed effort that Arild Ostbro pushed wide after a deflection.
Scotland were soon ahead after a superb team move. Rhodes took the ball in from Jack and laid it off to Scott Allan, who twisted and turned before rolling the ball to Cairney to pass the ball into the bottom corner from 16 yards.
Rhodes was unfortunate to be booked for diving by referee Padraigh Sutton, on the advice of one of the assistants, after being caught by a sliding challenge from Havard Nordtveit on the edge of the box.
Scotland remained in control with Dundee United midfielder Allan pulling the strings.
Gregg Wylde could have slipped Rhodes through but opted to shoot and could not get the ball round the defender before Allan pulled a left-foot shot after more good work from Jack and Rhodes.
Stark made several changes at half-time and in the early stages of the second half, which naturally affected the flow of the game, but his side remained comfortable.
Despite sitting deeper, they looked the likelier team to add to the scoring with substitute Leigh Griffiths turning inside a defender before shooting wide from a good position.
Rangers goalkeeper Grant Adam survived his first serious test around the hour mark, diving to save comfortably from Velon Berisha after a decent move.
And Scotland took advantage of some clinical breaks to move out of sight. Aberdeen right-back Jack curled high inside the far post from 16 yards after making space for himself in the 72nd minute.
And Dundee United midfielder Armstrong broke forward gracefully before running at heart of the Norwegian defence and shooting low into the bottom corner with his left foot from 22 yards three minutes later.