West Ham fought back from a goal down to draw at St Mary's, denying Saints a much needed win.
Rice's first league goal in more than two years came after West Ham failed to convert several chances.
While the Saints earned their first point in five games, a draw keeps the pressure on the manager.
Questions over Hasenhuttl's immediate future have arisen after a four-game losing run for Saints.
The first half here was bright for the Saints, with Che Adams denied twice by Lukasz Fabianski, but they didn't give up.
There was controversy over the opening goal as the West ham players claimed that referee Peter Bankes had stopped the Hammers attacker from getting to the ball ahead of Perraud.
The West ham fans appealed for a foul throw-in against Kyle Walker-Peters, but the visitors received no joy from the VAR.
But with the Saints unable to build on a promising first half and ultimately failing to keep their first clean sheet of the season, a draw will do little to improve the mood around the club.
West Ham were looking to extend their winning run to five games, having secured their place in the knockout stage of theEuropa Conference League with two games to spare.
During the first half, they were frustrated because they didn't work with the Saints keeper.
The first three minutes saw two shots sent to the wrong side of the post, one of which was missed.
The Hammers failed to score a first-half goal for the eighth time in 10 games this season, with Lucas Paqueta's headed effort hitting the outside of the post.
Rice popped up with a well-taken leveller to extend West ham's run and keep the Saints in the bottom half.
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