Joe Aribo and Kyle Walker-Peters scored within nine minutes of one another to cancel out Rodrigo’s second-half brace with Leeds poised to claim all three points.
Despite looking the most threatening of the two, Southampton failed to produce an end product, which is something that has plagued them for the last few seasons.
A sloppy start to the second half allowed Leeds to grab a 1-0 lead when Rodrigo turned in Jack Harrison’s cross one minute after the restart.
Rodrigo then doubled his tally as well as Leeds’ advantage with a header from close range on the hour mark.
Just when it looked like the visitors would spoil the home opener for Saints, the introduction of Aribo, Skeou Mara and Adam Armstrong and a change in shape held change their fortunes.
Aribo provided an instant impact as he pulled one back for Southampton in the 72nd minute, finishing off Armstrong’s cross from the centre of the box.
Walker-Peters then helped his side complete the comeback when he met Mara’s low cross with a low shot that beat Illan Meslier for pace into the bottom corner.
Southampton (3-4-2-1): Bazunu; Salisu, Bednarek (Mara 70), Kotchap; Djenepo (Stephens 90+1), Lavia, Ward-Prowse, Walker-Peters; Elyounoussi (A. Armstrong 61), S. Armstrong (Aribo 62); Adams
Unused subs: McCarthy (GK), Valery, Diallo, Romeu, Lyanco
Goals: Aribo (72), Walker-Peters (81)
Bookings: Salisu (78)
Leeds United (4-2-3-1): Meslier; Struijk, Llorente, Koch, Kristensen; Adams, Roca (Foreshaw); Harrison (Klich 84), Rodrigo, Aaronson; Bamford (James 28, Sinisterra 84)
Unused subs: Klaesson (GK), Drameh, Summerville, Hjelde, Greenwood
Goals: Rodrigo (46, 60)
Bookings: Harrison (43), Kristensen (45), Aaronson (90+3)