The best bet would be a small 555 timer circuit called a power-up one shot. This circuit can be found all over the web and gives a clean short duration output pulse when continuous power is applied to the 555 circuit. 555s can operate between 5-15 vdc and source or sink the output to trigger a small relay or transistor- based relay trigger.

The amp you are using evidently utilizes a flip-flop circuit internally to control its power supply so sustained current from an external source is not necessary.

This type of flip flop is extremely common.

Google 555 timer circuits and you will find hundreds of very simple apps.

I would hesitate to rely on the cap/relay circuit for long. The current draw on the cap by the current inrush of the relay will break down the cap and as the cap degrades, the current / voltage supplied to the coil will eventually cause the relay to chatter as the supply voltage decays. Depending on the input pulse time required by the amp, the chattering may actually toggle the amp on and off rapidly.