New signings Jack Hunt and Michael Mancienne started for the visitors, who also had trialist Sancidino Silva in action.
The newly-promoted Championship side scored the only goal of the game shortly after the hour-mark, when Ben Pringle provided an inch-perfect pass for Kieran Agard to control and finish past Reds goalkeeper Karl Darlow.
Eastick told the club's website: "There are a lot of good things to take out of the game.
"Some of our football up until the final third was very good, the whole back four did very well, there was some good rotation in the midfield and we played through the pitch very well.
"We're a shade unlucky not to get a draw but overall we need to be better in the final third. It's early days and this performance was certainly an improvement of the ones in Portugal."
Meanwhile, Forest have confirmed Henri Lansbury will be sidelined for around two weeks with an injury sustained in their recent match with Portuguese side Rio Ave.