Southampton were gifted a chance to take the lead as early as the fifth minute when Eric Bailly was adjudged to have handled the ball inside the penalty area, although replays suggested that the offence was committed just outside the box.
However, Gabbiadini - without a goal since March 4 - saw his spot kick superbly saved by Romero as he attempted to fire the ball into the bottom left corner.
On the quarter-hour mark, United hit back through Wayne Rooney, who played a delightful pass through to Martial but the Frenchman's shot was dragged wide of the post.
As the first half progressed, chances on goal dried up but just before the break, Redmond had a deflected shot from around 20 yards comfortably saved by Romero.
Southampton went back onto the offensive after the restart, with Dusan Tadic seeing a rasping shot from just outside of the penalty area punched clear by Romero, who was impressing as a replacement for David de Gea.
Soon afterwards, the stand-in goalkeeper saved brilliantly from Romeu's drive from 14 yards, before he made a third intervention in quick succession to tip James Ward-Prowse's curling effort wide of the post.
The opportunities continued to arrive with Ward-Prowse having a goalbound volley blocked by Phil Jones, and United's Martial curling against the post from 20 yards out.
With 10 minutes remaining, Jay Rodriguez became the next player to waste a good opportunity when he sent a shot from the edge of the area wide of Romero's post, despite having time and space.
That proved to be the final opening of a closely-fought contest, with the result potentially suiting both clubs ahead of their upcoming fixtures.