Following a goalless first half, Leicester took the lead in the 54th minute with a free-kick from James Maddison but super-sub Adams scored in the 68th and 84th minute to seal Southampton's first win of the season.
"What the manager has instilled in us is to never give up and keep believing," Adams said. "We did that today.
"First half was tough, in tough conditions, but we stuck in there and got the points."
Adams scored nine minutes after coming off the bench, finding the target with a low strike following a long throw-in, while the hosts were left to rue their poor defending.
Adams almost bagged a hat trick in stoppage time, but his shot rebounded off the post.
The miss, though, did not matter as Ralph Hasenhuttl's side picked up three points and moved up to 12th in the standings.
"You could see at the start we had some respect for playing here better than we can," Hasenhuttl said.
"After the goal, we see we have nothing to lose and the guys step up. We played some fantastic football.
Earlier, Maddison had put Leicester ahead after he whipped the ball around the Southampton wall from a free-kick and found the bottom corner of the net, marking his second goal in three games.
Leicester slipped to 18th and are yet to register a win this season after losing two matches and drawing the other.