Agnew was handed the reins by chairman Steve Gibson on Thursday following Aitor Karanka's departure, and he presided over an improved performance, but ultimately a 3-1 home defeat, against Manchester United on Sunday.
However, striker Gestede, who scored Boro's first goal in 510 minutes of league football after coming on as a substitute, believes he deserves the chance to see out the season at the very least.
He said: "I hope he stays until the end of the season now. Why not? He deserves his chance. He knows us, and I think it would be very hard for another manager to come in for the last 10 games. He is the right appointment.
"It has been a tough week for all the players. I will always be very grateful to him (Karanka) because he brought me into the club and gave me the opportunity to play in the Premier League again.
"Now, we are all behind Steve. We know he can bring some fire into the team because he has amazing spirit. He is trying to share that spirit with the team. He was here from the start of the season, so he knows us very well. I think it was natural for us to be under his management."
With 10 games remaining, Boro find themselves in 19th place and five points adrift of safety.
Perhaps more worryingly, they have not won in the league in 11 attempts dating back to December 17, and that has to change and change fast if they are to avoid the drop.
Gestede said: "We want to stay up, so we have to go for it and win games. That has to be the attitude no matter who we face.
"Manchester United are a good side, but we knew we had to play our way and that will be the same whoever we face in the rest of the season.
"We went with two strikers after I came on and made sure we put the ball in the box whenever we could because that is how you will score goals.
"We have no choice - if we want to stay up, we have to win games. That has to be our approach and we have to play to make sure that we score more goals than the other team. That has to be the only objective."