Newell must now fear for his job after the Saints withstood some late pressure to win comfortably.
Southampton have now gone five matches without defeat and have manoeuvred themselves into a strong position on the fringes of the play-off positions.
The Saints were much the stronger side in the first half and should have been well ahead by the interval.
But Luton came back strongly after the break, pulled a goal back through Russell Perrett and came close to snatching an equaliser.
Luton had the game's first real opportunity in the 11th minute after Ahmet Brkovic had set up a chance which Warren Feemey turned against the post from close range.
Luton had on-loan goalkeeper Dean Kiely in goal and the Portsmouth player produced some important saves over the afternoon.
But after 14 minutes he could have done better when Chris Baird headed goalwards from a centre by Gareth Bale. Kiely got his hands to the ball but could not prevent it falling behind him into the net.
It was a sweet moment for Northern Ireland international defender Baird because his goal was his first in four years and 50 appearances for the club.
Kiely was lucky in the 18th minute when he carried a shot from Grzegorz Rasiak but was redeemed by an offside flag.
Southampton went further in front in the 36th minute when Leon Barnett handled under pressure from Rasiak and the Polish striker got up to score from the penalty spot for his 11th of the season.
Kiely kept Luton in the match with saves from Rasiak and Rudi Skacel before Luton surged back into contention with a goal in the 66th minute.
Lewis Emanuel took a free-kick on the left and, as former Luton goalkeeper Kelvin Davis advanced from his line, Hampshire-born Perrett headed over him and into the net.
Kiely once more came to Luton's rescue when he tipped over a Skacel free-kick from long-distance before the Hatters mustered one frantic last attempt at an equaliser in the three minutes added on for injuries.
Luckily for Southampton Portuguese midfield player Pele came to the rescue with a saving header as Feeney threatened to finish off another Emanuel cross.