Despite losing Marten de Roon to an ankle injury and seeing Gaston Ramirez sent off in the 20th minute for a second yellow card as they fell nine points from safety, Agnew is adamant they have not given up.
Their performance at the Vitality Stadium, in which they were uncharacteristically poor in defence, left them in 19th, and their five remaining fixtures are against Sunderland, Manchester City, Chelsea, Southampton and Liverpool.
Agnew, who has overseen four defeats and two draws since his appointment, said: “It’s very emotional. There’s lots of angry players, lots of emotional players, as there is staff. We have to be realistic. It’s a huge, huge challenge, but all we can do is dust ourselves down and get ready for Sunderland (on Wednesday). Every game we need a win. We remain as positive as we can be.
“In many ways we lost the game in the first 10 minutes. That’s not like us. It was really, really poor defending.”
Regarding the shock death on Friday of club favourite Ugo Ehiogu, he added: “What a sad, sad day. Our condolences go to the family; the supporters were amazing again.”
Two first-half goals from Joshua King and Benik Afobe and others in the second from Marc Pugh and Charlie Daniels, lifted Bournemouth to 12th – seven points above the bottom three.
AFC Bournemouth: Boruc, Adam Smith, Francis, Steve Cook, Daniels, Fraser (Stanislas 72), Arter, Gosling (Lewis Cook 44), Pugh, King (Mousset 84), Afobe. Unused substitutes: Gradel, Allsop, Mings, Cargill.
Middlesbrough: Guzan, Barragan (Forshaw 23), Chambers, Ayala, Gibson, Friend, Ramirez, de Roon (Da Silva 38), Clayton, Downing, Negredo (Gestede 80). Unused substitutes: Konstantopoulos, Bernardo, Stuani, Traore.
Referee: G Scott (Oxfordshire).