Pierre-Emile Hojbjerg scored a sensational long-range drive and Southampton were seemingly on track to record successive league wins when Danny Ings converted a penalty he won himself.
But Shane Duffy's header put Brighton back in the game and when referee Anthony Taylor penalised James Ward-Prowse for holding the defender at a corner, the stage was set for Murray.
The veteran striker struck the ball down the middle to beat the diving Alex McCarthy, taking his tally to five goals in six league starts for Brighton against the Saints.
Southampton dominated the opening exchanges, Anthony Knockaert scrambling an Ings effort off the line and Mohamed Elyounoussi wasting a good headed chance.
And a fantastic strike from Hojbjerg gave the hosts a deserved 35th-minute lead.
Lewis Dunk's poor clearing header went straight to the midfielder, who let fly with a 35-yard drive that dramatically swerved into the bottom right corner of Mat Ryan's net, leaving the visiting goalkeeper helpless.
Shane Long wasted a one-on-one opportunity after the break before Ings was bundled over by Gaetan Bong in the box, allowing him to roll home Southampton's second from the spot.
Their two-goal advantage lasted just two minutes, though, as Knockaert's curling free-kick allowed an unchallenged Duffy to glance home a header.
Substitute Alireza Jahanbakhsh struck the outside of the post in the 80th minute but Murray, so often Southampton's scourge, earned his team a point with a penalty for the second game in a row after a 2-2 draw at Fulham before the international break.
Taylor pointed to the spot when Duffy was sent sprawling in the box, with Murray gleefully converting in the first minute of added time to ensure the St Mary's spoils were shared.
What does it mean: Southampton's soft centre
Southampton were in serious trouble when Mark Hughes was appointed in March but he steered the club to safety relatively easily. The signing of Ings from Liverpool has given them a much-needed goal threat but ongoing defensive weaknesses were exposed late in the game, meaning the former Stoke City manager is still waiting to record back-to-back Premier League wins for the first time since January 2017.
Pat on the back: Hojbjerg adding goals
Having found the net in a 2-0 win at Crystal Palace last time out, Hojbjerg was on target again in spectacular fashion. Scoring two goals in two matches equals his haul from his previous 102 appearances in Europe's top-five leagues - and his memorable effort deserved to be in a win.
Boot up the backside: Long miss costly
Brighton came out of the blocks brilliantly after the break, dominating the first 10 minutes of the second half. Long had a great chance to score when sent through by Ings, but the Republic of Ireland international unable to beat Ryan. Although Ings later extended Southampton's lead from the spot, an earlier second goal could have settled the contest.
Key Opta stats:
2 - Danny Ings has scored in consecutive Premier League appearances for the first time since February 2015. Revitalised.— OptaJoe (@OptaJoe) September 17, 2018
- Southampton are winless in their last 11 Premier League games played on a Monday (W0 D4 L7).
- Brighton have managed just two away wins in the Premier League since the start of last season, fewer than any other side to appear in both the 17/18 and 18/19 editions (W2 D5 L14).
- Knockaert has registered three assists in his last four Premier League games, after managing just one in his previous 43 games in the competition.
- Southampton have dropped 26 points from leading positions in the Premier League since the start of last season, the most of any side to appear in both campaigns.
- Hojbjerg's strike for Southampton came from 34.9 yards out, the longest-range strike in the Premier League this season.
Southampton face a tough away assignment with a trip to Liverpool on Saturday, while Brighton host Tottenham in the same day's late kick-off.