A game of little genuine quality was settled four minutes into the second half when midfielder Craig Gardner latched on to a loose ball 22 yards out and smashed it past keeper Kelvin Davis off the underside of the crossbar.
However, Southampton, who were below-par throughout, could have snatched a draw at the death with Adam Lallana and Rickie Lambert both failing to make the most of driven Nathaniel Clyne crosses, with Lambert firing over an open goal from just three yards.
Poyet made nine changes to the team which lost to Hull last weekend but saw his understudies get the better of a Southampton side which included only five of the men who started against the Tigers on Tuesday evening.
Assistant manager Jesus Perez, standing in for Mauricio Pochettino at the post-match press conference, was quick to defend the decision not to field a stronger side.
Perez: "Always, we do the same. After the last game, we assessed our players and we decided the best team to play today.
"Obviously, it was a tough game against Hull, we played away as well, we came home late and some of our players were not fully recovered and we decided the best players to play today.
"For us, every single player is important and useful for us and it doesn't matter who is on the pitch.
"When we win, we win as a team. Today, we didn't get anything as a team as well.
"But I have nothing to say. There was a lack of freshness, that's obvious, but nothing more.
"When we decide to play with one or another player, it's because we trust them, they trust us. They support us, so there's nothing to complain about."
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