After this first – and crucial – league away win over Manchester United since May 1968, a beaming Gus Poyet lauded the "big cojones" of his Sunderland side.

No one can argue with the intestinal fortitude of a band of players who have all but averted relegation, despite propping up the Premier League on 15 April with 25 points from 32 matches. That was the result of a dire run after the 3-1 Capital One Cup final defeat by Manchester City on 2 March, in which Poyet's team lost five times in six league outings until they rolled up at the Etihad Stadium midway through last month.

With all the apparently smart money on Sunderland already being sunk, they managed a fine 2-2 draw against Manuel Pellegrini's side. Chelsea were subsequently beaten 2-1 at Stamford Bridge to give José Mourinho a first league defeat there in his 78th home game, and Cardiff City were routed 4-0 at the Stadium of Light on 27 April. On Saturday Sebastian Larsson's 30th-minute volley proved enough to give Ryan Giggs a first defeat in his second match as interim manager.

All of this was described as a "miracle" by Poyet. Now, Sunderland are the team no side – including United – want to face. "It is nice. We were talking about whether we can keep playing for a couple more months," the manager said. "I think that was the idea from the beginning. I thought, from the beginning, that after January that would be the case. But everybody knows after the final we went on a run that was scary. It was not easy, I can tell you that it was not enjoyable at all. You need to believe, you need to make sure everyone believes and keeps doing it.

"If you give your all and do your best then sometimes you are not good enough and you need to take it. We did not want to give up and especially on the players on the pitch. They deserve the credit because it's the way they played."

Poyet stressed how his team kept going in the final stages against United. "You are really tired with 20 minutes to go, then you see [Danny] Welbeck and [Robin] van Persie coming on. You need to be strong. You need to be strong and have someone special to keep defending and make sure they keep a clean sheet here after the game last weekend. They were back to the Sir Alex [Ferguson] times [in the 4-0 win over Norwich City] because they were impressive," he said.

The equation is that after Norwich's 0-0 draw with Chelsea, Sunderland's superior goal difference of 13 means Poyet's side effectively need only a point from their final two matches – at home against West Bromwich Albion on Wednesday evening and Swansea City next Sunday.

Connor Wickham created Larsson's strike with a neat turn and cross, although Darren Fletcher should have been closer to the Sunderland No10 and Michael Carrick was guilty of the same crime regarding the Swede, who was allowed space to finish beyond David de Gea.

Of his squad's mentality, Poyet said: "We say in Spanish have the group and have the group go together. They commit. The commitment of the players together to do something special. It is the key. The belief, the commitment, the fighting spirit, the not accepting what people had taken for granted and to keep going. That is incredible. Big cojones, you need."

Larsson said of the battle against relegation: "We realised it was going to be very tough but we never felt we were dead and buried. We have stuck together and we believed we could do it somehow. We have to stay focused and make sure we get over the line. It would be a terrific turnaround if it does happen.

"If we do end up staying up it will be one of the proudest moments for me in my career. If you think of everything that has gone on, I will be really proud."

Giggs could have Wayne Rooney back for Tuesday evening's visit of Hull City after he missed Saturday's game because of a tight groin and a stomach bug but the interim manager offered no excuses about his team's lack of quality against Sunderland. He still wants United to try to catch Tottenham Hotspur, beaten at West Ham United, for a Europa League place.

"I still believe that United should be in Europe and, yeah, it was a chance missed – Tottenham losing. We could have staked a claim for sixth place, so that's disappointing as well."

Man of match Connor Wickham (Sunderland)