Hi,
charging a capacitor from the output of a (?CMOS) device is proably not sufficient.
Better charge the capacitor after rectification of the output signal of the buzzer circuit and use a transistor to switch the relay.
I simulated the buzzer circuit using an astable built with a timer IC. With a cap of 47µF and a parallel resistor of 1MΩ the relay stays activated for about 1 second after the buzzer signal returns off.
To modify your circuit just use the parts connected to and after pin3 of the timer IC:
D2..D4, C1, R1, R5 and Q1. R4 and D1 are optional.
See the attachments, also showing the voltage incraese after rectification.
Boncuk