Derby had a couple of chances to score even before Christian Dailly gave them an early lead after eight minutes. However, within five minutes Gordon Watson and Matthew Oakley had put Southampton ahead, a lead which Jim Magilton increased late in the game.
Smith was particularly critical of his goalkeeper Russell Hoult: "There were other things that went wrong, but in the final analysis the guy is there to stop it going in. The first shot should have been handled better and the cross [for the second goal] should have been caught. He didn't thrill me at all." He added: "When we went behind we seemed to think route one would be our salvation and never got our passing game going."
Aljosa Asanovic added to Smith's problems when he gave a show of petulance after being substituted in the second half. Smith said: "He should be used to it. We sub him every game. He was probably disappointed with his own performance."
To his credit, Smith refused to blame the flu epidemic which hit the club last week and cost him Paul McGrath and Dean Sturridge for this game.
For Southampton, this was a fine return to form after a worrying run of defeats. Even without the injured Matthew Le Tissier, who is expected to return for the Boxing Day game at Tottenham, Southampton were good value for their win.
Graeme Souness, their manager, said: "It's good to get a win after five losses and hopefully that will be the start of what will be a good Christmas for us.
"It was most important for us that we got the taste of winning again. It's half-way through the season and we know what we have to do. We need spirited performances to keep us in the League. There's a long way to go."
Goals: Dailly (8) 0-1; Watson (10) 1-1; Oakley (13) 2-1; Magilton pen (90) 3-1.
Southampton (3-4-1-2): Beasant; Lundekvam, Maddison, Dryden (Magilton, 8); Oakley (Hughes, 85), Benali, Slater, Charlton; Berkovic; Ostenstad, Watson (Basham, 89). Substitutes not used: Potter, Moss (gk).
Derby County (4-4-2): Hoult; Rowett, Stimac, Yates (Gabbiadini, 45), C Powell; Flynn, D Powell, Asanovic (Willems, 73), Laursen (Carsley, h/t); Dailly, Ward. Substitutes not used: Carbon, Taylor (gk).
Referee: P Danson (Leicester).
Bookings: Southampton: Benali, Slater. Derby: Gabbiadini, Rowett.
Man of the match: Berkovic. Attendance: 14,901.Reuse content