Defender Christophe Berra headed in a Leigh Griffiths corner in the ninth minute of a flat and insipid first-half before the Celtic striker added a second from close range four minutes after the restart.
The two-goal cushion allowed attention to turn firmly to Wembley, where Group F leaders England came from behind to beat second-place Slovakia 2-1.
With two fixtures remaining, Gordon Strachan's side moved on to 14 points, one behind the Slovaks who visit Glasgow next month before Scotland travel to Slovenia for the final fixture. Two wins would secure second spot and a play-off place.
Scotland, who stretched their unbeaten run in the section to four games, will have to play a lot better in their final two matches to keep hopes of reaching Russia alive but at least those hopes remain intact.
There was renewed optimism among the home fans following the 3-0 win in Lithuania on Friday night with Strachan fielding an unchanged side, with only left-back Andy Robertson having also started Scotland's 5-1 win in Malta a year ago.
Hampden Park, however, was only around half-full with a rather eerie atmosphere initially for a game against the section makeweights who had no points from seven games.
Griffiths' corner had led to Armstrong's opener in Vilnius and a similar delivery from the right saw Hearts defender Berra climb above Samuel Magri to head in from six yards.
The goal should have allowed the Scots' players and fans to relax but there remained an underlying anxiety.
Armstrong hit the side netting after bursting into the box and Griffiths drilled a shot from distance wide but it was all a bit stodgy.
News of Slovakia's goal against England numbed the home fans until the Three Lions' leveller brought them back to life, but the first half remained low-key.
There was a hint of offside about Griffiths' goal just after the break.
Armstrong played in Robertson whose cut-back was turned against the post by James Morrison, on for McArthur at half-time, with Parkhead attacker Griffiths left with the easiest of tap-ins from four yards.
Griffiths could have grabbed his brace in the 67th minute when he was sent clear by Phillips but his weak shot was easily saved by Hogg, before the Scotland forward immediately came off with a calf injury, replaced by the burly Chris Martin.
The closing stages saw Phillips blaze a shot high over the bar but the points were already safe and the Tartan Army looking forward to the visit of Slovakia next month.
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