Billy Stark's side struggled to break down a stubborn Luxembourg side in the first half but stepped up the tempo after the break and Stuart Armstrong and Leigh Griffiths netted within five minutes of each other.
Substitute Tony Watt scored a late penalty and Scotland now travel to Austria looking for the 16-point total they are expected to need for a play-off spot.
The victory leaves unbeaten Scotland three points behind Group 10 leaders Holland and four clear in second place.
Scotland need Austria to beat Holland in Tilburg on Friday night in order to have any chance of topping the group, although they are seven goals worse off than the Dutch.
Anything but an Austria win would guarantee Scotland second place but Stark believes they still need to win on Monday to be one of the four best runners-up that qualify for the play-offs.
Stark has seen James Forrest and Jordan Rhodes, scorer of eight goals in the previous eight games, promoted to the senior squad since the campaign began and another of his creative forces, Dundee United winger Gary Mackay-Steven, was missing through injury.
That prompted a change from the usual lone striker formation to a 4-4-2 and Scotland struggled to carve open the visitors in the first half.
Luxembourg pressed the ball aggressively from the start but captain Paul Hanlon created an early chance when his pass found Johnny Russell in the left channel, but the Dundee United striker's shot from a tight angle was pushed wide.
Luxembourg were shutting out their hosts and they almost took the lead in the 25th minute. Tottenham goalkeeper Jordan Archer, making his first start amid injuries to Mark Ridgers and Grant Adam, dived full length to push wide Billy Bernard's header following a free-kick from the right.
Scotland were struggling to get the ball into the box, though Russell saw a long-range effort comfortably held before Danny Wilson curled over after a corner was half-cleared.
Stark's team managed to push men forward in the closing stages of the half but were still lacking a cutting edge.
Hanlon missed a great chance four minutes after the break when he blazed over from close range after a Griffiths free-kick had been parried by goalkeeper Philippe Hahm.
Scott Allan had won the free-kick with a forward burst and the West Brom midfielder looked increasingly like Scotland's best chance of creating a breakthrough.
More skill from Allan created space for Russell on the left and he crossed for Griffiths, but the forward went for the spectacular scissors kick when a header might have been on and he failed to connect properly.
It was perseverance that finally broke down the visitors in the 63rd minute. The away defence closed down Griffiths after a one-two with Allan but Russell ran in and tried a first-time shot that broke off a defender and spun across the 18-yard line.
The ball fell kindly for his Dundee United team-mate Armstrong, who curled the ball into the roof of the net.
Stark immediately brought on right-winger Peter Pawlett and Russell moved out to the left to form a more familiar 4-5-1 formation.
Scotland soon doubled their lead when Russell broke down the left and cut the ball back for Griffiths to sweep home from eight yards.
Luxembourg had a penalty claim denied when Icelandic referee Thorvaldur Arnason ruled Wilson got the ball in a risky slide tackle on Tom Laterza.
Arnason pointed to the spot at the other end when Watt's shirt was pulled just inside the box and the Celtic striker sent Hahm the wrong way in the 83rd minute.