Dick Advocaat is confident Sunderland can keep themselves up after seeing them rekindle their Barclays Premier League survival bid with victory over Southampton.

The delighted Dutchman looked on as Jordi Gomez converted two penalties to secure a 2-1 win at the Stadium of Light and three points which proved vital on an afternoon when Leicester and Aston Villa also boosted their chances of escaping the drop.

Sunderland remain in the bottom three but just two points adrift of 14th-placed Villa, and asked if there was enough in the dressing room to get the club out of trouble, Advocaat replied with a smile: "Yes, definitely – I'm there. And 18 players.

"We have to believe in it. What is it – five games we have played now? Seven points? There are still four to go.

"Everything is still open. We have still everything in our own hands, and that's the most important thing."

The home side took a 21st-minute lead when Gomez dispatched his first spot-kick after Jose Fonte had caught striker Danny Graham in the face with a flailing boot, but squandered it within seconds as keeper Costel Pantilimon collided with defenders Sebastian Coates and John O'Shea to present Sadio Mane with the easiest of chances to level.

Gomez was handed a second opportunity from 12 yards 10 minutes after the break when James Ward-Prowse bundled Jermain Defoe to the ground and was promptly sent off, and he accepted it with equal composure.

The visitors belatedly responded and might have snatched a point four minutes into stoppage time when Pantilimon redeemed himself with a fine reaction save from substitute Steven Davis' shot.

Advocaat said: "That's the reason he is a goalkeeper. He had to save that ball after the first goal.

"Knowing the results from the other teams, it was really necessary to win. Today, it was a real team performance. They worked so hard to get this result and in the end, I think we also deserved it.

"We created enough chances to make the decision quicker but against a team like Southampton with very good players, they can always score out of nothing.

"They worked so hard, the players. But I was really happy when the final whistle came."

Advocaat initially set his players a minimum target of six or seven points to save themselves, but having managed seven from his first five games he is well aware that is far from enough.

He joked: "We have got seven points, so I can go.

"That's what I said, if we get six or seven points, that will be enough. But it's not enough, unfortunately."

It proved a sobering afternoon on Wearside for Southampton boss Ronald Koeman as his hopes of securing European football were dealt a blow.

Koeman said: "We didn't reach our level, and I'm frustrated about that. But it's strange that with one player less on the pitch, we showed our football and we showed our qualities, and before not.

"We made too many mistakes in our ball position. The first 10 minutes after half-time was very poor and out of that situation came one of the penalties and the red card, and then you know it's difficult.

"Then with one player less, we were the better team and the dominant team, and that's difficult to understand.

"In my opinion, it's not about spirit, about ambition. My criticism to the players is not about what they did and what they tried to do. My criticism today is about our level in our ball position. Normally it's much better than today, and that was the reason and not the referee.

"Sunderland put everything in – that's normal in the situation they are in – but I expect much more of the team than we saw today."

Mobile app users, watch the video highlights.