Simple answer: yes
Your rocket would need to lift off on the side of the planet facing the gas giant. If you launch from the other side you're fighting the gravity of both planets.
Having said that, here's the simple lowdown.
If the gas giant is close enough that its gravity overwhelms Earth's — meaning that "down" to some small degree is in the direction of the gas giant, then you'd need almost no fuel at all. But this also means that everything is being pulled toward the gas giant, making navigation a blooming nightmare if not impossible.
If the gas giant is close enough to negate Earth's gravity then things on Earth start to float. Basically the same benefits of overwhelming Earth's gravity without as much navigational debris.
With every meter separation after those first two conditions, your fuel requirements increase until you get to the normal escape velocity.
So, the trivial answer to your question is "sure, you can use it to boost a launch."
Complex answer: when the gravity boost comes around, it's probably too late to matter
@John's also right. The closer you get to Earth, the more destructive the situation becomes. If a gas giant's gravitational field is strong enough to significantly counter Earth's own gravity, you're deep into wreck-the-planet mode. You're not just pulling up water, you're pulling up everything else, too, which means substantial vulcanism and a noticeable wobble to Earth's rotation due to all the water being pulled to the side of the planet facing the gas giant.
But, if "realism" is to rule the day (I'm not a fan...), then realistically, by the time you wait for a substantial fuel reduction due to the gravitational influence of the gas giant, the rocket is either already destroyed or can't get away from Earth due to all the chaos.
And just to add insult to injury...
Remember when I said the simple answer is "yes?" Well... if you save the fuel getting off Earth, you still need a boat load of fuel to escape the gas giant's gravity well. One could say that a launched rocket could try to skirt around the gas giant, even perhaps using it as a sling shot... but if it's close enough to nullify or all but nullify Earth's gravity, then you're too deep into the gas giant's gravity well to do that. In other words, "realistically," even if you got off the planet cheaply, you'll burn more escaping the gas giant than you ever would simply launching before the gas giant grew close enough to affect the Earth.
But does that matter?
How "realistic" do you need to be? Take that simple yes and run with suspension of disbelief.