Southampton were comfortably the better side in the first half and despite a change in approach from Luton after the break which saw Town have two Drew Talbot efforts disallowed for offside, the hosts had little to show for their efforts.
The result leaves Luton rooted to the bottom of the Championship, while for the visitors, the points boosted their hopes of promotion.
Town stuck with the adaptable 4-5-1/4-3-3 formation used at Burnley in their last outing with Kevin Foley playing the holding midfield role and Ahmet Brkovic and Dean Morgan the wide men asked to support lone front man Talbot.
The system offered promise early on as Morgan found Brkovic, whose cross from the right almost fell into the path of the onrushing Matthew Spring in the second minute.
Luton harried and chased and tried to get their midfield involved where possible, but it was Southampton who created the chances.
Gareth Bale's run and pass allowed Jermaine Wright to fire just over from the edge of the penalty area before Lewis Emanuel almost deflected Andrew Surman's 24th minute cross into his own net only for the ball to bounce into the turf and over the crossbar.
Saints continued to press and after forcing two corners in quick succession, Viafara saw his deflected 16-yard shot pounced upon by Saganowski, who spun and hit a six-yard effort against the outside of the post and wide.
Danny Guthrie tested Dean Brill with a low 15-yard effort in the 41st minute before the visitors opened the scoring three minutes later as Surman and Guthrie combined to feed Viafara, who held off Chris Coyne before hitting a low left-foot shot into the bottom corner from 14 yards.
Not content with that, Southampton doubled their advantage in first-half injury time when goalkeeper Bartosz Bialkowski's long clearance was flicked on by Leon Best for Saganowski to smash home an unstoppable volley from 20 yards.
Luton reshuffled at the break bringing on Calvin Andrew for Morgan and Besian Idrizaj for Foley and the move almost had immediate impact as Brkovic's 48th minute corner was helped on by Spring and Andrew for Talbot to convert only for the effort to be disallowed for offside.
The visitors then had an effort of their own chalked off in the 54th minute when Saganowski nodded in Bale's free-kick only for Luton to be awarded a free-kick.
Incredibly, the Hatters then had another effort disallowed in identical circumstances to their first as this time Brkovic's 59th minute corner was guided on by Spring and then Idrizaj for Talbot to net again only for the former Sheffield Wednesday man to be again flagged offside.
Southampton continued to threaten, while Luton's day was summed up when Idrizaj flicked on for Brkovic to rifle a 75th minute effort that looked bound for the back of the net only to bounce back off the far post.