Agnew was installed as the club's interim head coach on Thursday after Aitor Karanka's reign on Teesside drew to a close after almost three and a half years.
As he takes up the reins, Boro lie in 19th place in the league, three points adrift of safety and without a win in 10 outings, but the former Leicester and Sunderland midfielder is confident he can pull off a rescue mission.
Agnew, who has drafted in the hugely experienced Joe Jordan as his number two, said: "It would be the biggest single achievement of my life. I sit here and I keep using the word 'confident', and I do believe we will stay in the Premier League.
"I think I'm capable of delivering it, the players are capable of delivering it and I know we have got the support of everyone else in the club, and the supporters will play a massive part in that."
Asked if there is enough in the Boro camp to engineer an escape with rumours of splits in the dressing room being rife in recent weeks, he said: "Absolutely, yes.
"The players understand where we are and what we need to do to get out of the situation we're in. But they also realise that we've played some really, really good football, we've played well against a lot of the top teams, so they are more than capable."