Scott Parkers side felt they should have had a penalty after a ball from Lookman was flicked onto Jan Bednarek’s arm, which was close to his body, but referee Darren England allowed the game to continue and VAR affirmed the referee’s decision on review.
Despite struggling to make an impact early on, James Ward-Prowse almost gave Southampton the lead in typical fashion, with his 25-yard free-kick rebounding off the woodwork before Che Adams missed the rebound.
Shane Long and Theo Walcott both looked to have broken the deadlock for the visitors in the second half, only to have their efforts ruled out for offside on review as Saints slipped to eighth in the table.
The Cottagers played out their fourth consecutive draw and did so in the absence of their manager Scott Parker, who remains in self-isolation after a member of his household tested positive for coronavirus.
There were just three shots on target during the game, with possession also evenly shared between the sides.
Southampton had previously gone on a seven-match unbeaten run earlier in the season, during which they briefly topped the table, but the result sees them drop out of the European places.