Keizer was only appointed in the summer as a replacement for Petr Bosz, who left for Borussia Dortmund and has also since been sacked by the German side.
The 48-year-old has taken Ajax to second in the Eredivisie table, but an early exit from the domestic cup, as well as failure to qualify for both the Champions League and Europa League, has cost him his job.
Bergkamp and Spijkerman, working in an assistant manager capacity, have also been dismissed with immediate effect.
"The club management does not have enough confidence that the ambitions of Ajax will be realised with them," an official club statement read.
"The disappointing sporting results are also the reason for this decision. With Dennis Bergkamp, there is also a difference of opinion about the technical policy to be implemented."
Former Netherlands defender Michael Reiziger will now take over on an interim basis.