Jordy Brouwer and Besian Idrizaj both spurned good chances while Nabil El Zhar hit the woodwork and the misses proved costly when the Dutch side scored from a set piece.
"It is disappointing to lose the game because I felt we were the better side," reserve team boss Gary Ablett told the club's official website.
"We felt we should have had a penalty when Bes was fouled but the referee turned it down. The winning goal for Ajax came from a set piece that we defended badly and we will need to work on that in the next training session.
"The lads fitness levels have been good and we have been playing some good attacking football. The only problem has been not taking our chances when we get them, but overall I have been happy with the players in pre-season."