Norwich were punished for sitting back in the 36th minute as Scottish international Adams came out on top in a goalmouth scramble, scooping the ball home at the second attempt for his seventh Premier League goal of the season.
Having dominated possession, Southampton should have been out of sight by the break, but Mohamed Elyounoussi wasted two great chances, first sending a volley into the ground that bounced up straight at the keeper and then dragging a shot wide just before half-time.
Adams almost scored again for the home side four minutes after the break from another set piece, but his header was deflected out for another corner.
Norwich struggled throughout to maintain possession or mount any cohesive attacks, and though a few rare forays forward resulted in set pieces, they did little to threaten from them.
The visitors did not manage to register their first shot on target until deep into the second half and when Kyle Walker-Peters volleyed straight at the keeper, a second goal for Southampton seemed almost inevitable.
Elyounoussi wasted another great opportunity in the 87th minute, but the second goal finally came a minute later as Romeu connected perfectly with a loose ball to fire home and inflict another dismal defeat on the visitors.
"Norwich didn't have any player on the edge of the box so I knew we'd have chances. We needed that second goal, we were struggling a bit but that gave us the confidence," Romeu told Sky Sports.
Norwich remain rooted to the bottom with 17 points after 26 games. Southampton climbed above Brighton & Hove Albion to ninth spot with 35 points after winning for the third time in their last four games.
"We've always worked hard, our pressing is one of the best in the league and when we get that right not many teams can play through it," Adams told Sky Sports.
"We all had an agreement before the season to improve on last season. We have to take it game by game and see where we end up."