I imagine that +1GHz signal generator is quite nice, but it's frequency response is not flat. Without a signal generator, obtaining the frequency response of your filter WON't be quick and painless, but it IS possible. At its most basic implementation, you could just put the signal generator's output to your DUT's input, and measure the DUT's output with your SA at a whole bunch of frequencies and plot it.
If you just want some basic values like the -3db dropoff etc, then you could just measure the amplitude until the voltage gets to be -3db or 0.7071 times the voltage. You can play around with the sweep function (if it's present on your generator) and different settings in the SA, but DO NOT expect your external sweep generator to act as a tracking generator; the signal needs to be synchronized to the scan rate of the SA for that to work.