David Moyes believes Manchester United are dark horses to win the Champions League after the 3-0 victory over Olympiakos that put them through to Friday's draw for the quarter-finals.

The United manager went into the second leg of the last-16 tie – with his side trailing 2-0 from the first game in Athens – under pressure over his job because of the club's poor season. But Moyes gained much needed respite on a memorable night at Old Trafford thanks to a hat-trick from Robin van Persie, whose late injury he played down.

It was the first time United had overturned a two-goal deficit in Europe for 30 years and, even though they are least favoured of the quarter-finalists with the bookmakers, Moyes was optimistic about their chances.

"I suppose we will be underdogs in the draw, but I genuinely believe this club is capable of winning it," he said. "If we show our capabilities, which we haven't done too often, we can be a match for any team. After Sunday's performance [the 3-0 home defeat by Liverpool], we needed to bounce back and we did it in style with this victory. It was a really good result in Europe, we came from two goals behind. It means we have got ourselves another two games in the Champions League, something we will really look forward to."

Van Persie scored twice in the first half, the opening goal a penalty, and added his third from a free-kick in the 52nd minute, and Moyes said: "To score a hat-trick in Champions League football is a big thing, only certain players in the world are capable of it and he is one of them. He took his penalty and the free-kick well and Wayne [Rooney] did really well for the second goal [with the final pass]. He scored his hat-trick and really deserved it."

The Dutchman's injury, which he picked up towards the end of the tie after being challenged by Kostas Manolas, might have taken the gloss off the evening. However, though Van Persie had to be taken off on a stretcher, Moyes said: "I don't think it's too bad, a knock behind his knee. We will assess it in the morning. We are hopeful that it isn't too bad."

Despite the impressive performance and the result, Moyes did not consider the night marked a watershed in the club's fortunes under him. "We hadn't played well in the first game – it was only another game, but I wouldn't use the word you used, that's for sure," he said.

Yet with his side away to West Ham United on Saturday and then at home to Manchester City on Tuesday, Moyes believed the result could have a galvanising effect. "I've said it before and it hasn't ended up as we wanted, I don't want to say this is the moment, but in the same breath I really hope it is. It is a work in progress and it will take time to get it exactly how we want it," he said.

The 40-year-old Ryan Giggs, making his first start since 28 January, gave a vintage performance. His pass to Van Persie led to the penalty and he produced the ball from which Rooney created his strike partner's second goal. "Ryan's two passes made two of the goals, his all-round play was really good," Moyes said. "He defies his age and is a great player. Anybody who has been at United knows he is a great player."

Asked why Giggs had been limited to only 12 starts before Wednesday night, Moyes explained: "I thought that he is 40 years old and I needed to look to the future, to give people opportunities to play. But I felt after the result [on Sunday] and tactically, I needed him to carry out a certain role which he knew and did perfectly as well."

Moyes dismissed reports that the club's owners were considering his position. "I'm not feeling any pressure from inside the club, it's coming from you people [the media]. We know the job we have to do, it's probably bigger than what I thought when I first came to the club," he said. "There are stories of this player falling out, or that one falling out, but it's rubbish. People are looking for reasons why we are not doing well but it is only because we have not been playing well."

Rooney said the players had owed Moyes a victory. "We had a desire to win, and as you can hear from the fans now and against Liverpool, they were fantastic and they deserved this. We owed ourselves [a result like this] and the fans, the manager, everyone. This a great result and hopefully it can kickstart our season."