Short answer - Yes, according to theory anyway...
But more importantly, how?
Brief context explanation as you might be wondering what question could possibly lead to the conclusion that Saturn saved us from Jupiter?! As in, why would we even ask ourselves whether that's true or not.
Thousands of telescopes from all around the world are peering into the deepest regions of the observable universe as we speak, and the questions that we ask ourselves almost always stem from these observations.
Something that's become apparent to astronomers is that a very large proportion of the solar systems that we observe contain a planet that's larger than even Jupiter in mass and volume, which orbits around their relative star even closer than Mercury does to the Sun!
A Superheated gas giant.
Obviously, our solar system doesn't contain one of these (Jupiter orbits the Sun around 500 million miles away), so naturally we've asked ourselves why we're different.
Now here's the part that you're actually here for - the theory....
The Nice Model
This particular theory is known as the Nice model, and its underlying principles are that the 4 gas giants (Jupiter, Saturn, Uranus and Neptune) formed much closer to the Sun than where they can be observed today - somehow, they migrated outward.
While the Sun was in its early formation, nearby gases were being superheated by friction and pulled in towards the Sun by its strong gravitational field, but debris such as rocks, ice and gases further out from the Sun gradually started to clump together, as they collided with one another.
When enough of these collisions happen to form a body 1km in diameter, they're referred to as planetesimals (think baby planets).
This would've been happening all over the regions of the solar system where it was cool enough, but eventually, as they collided together to form even bigger bodies, there ended up being several distinctively larger planetesimals which, due to an increase in mass, started to gain ultimate gravitational control of nearby debris, allowing them to grow even larger.
The first planetesimal to achieve this status was Jupiter. and as it began to swallow up more and more mass in the form of debris left over from the Sun's early formation, its momentum started to slow, of which there's 1 mechanism to blame for - conservation of momentum.
Fundamentally, this principle suggests that momentum, because it's indirectly a measure of energy (kinetic), it cannot be created and it cannot be destroyed - it can only be transferred between entities.
In the case of Jupiter, as it sling shot matter out behind and away from it, because that matter gained momentum (it gained velocity due to being accelerated by Jupiter's gravity), then Jupiter must lose momentum.
Losing momentum is bad news for planets though, as it means that they begin to slow down, gradually shrinking their orbits such that they fall closer and closer to the parent body, which is in this case the Sun.
Everything in our solar system is falling towards the Sun, but their velocities result in them constantly "missing" the Sun and being slingshot around to form their orbits.
As soon as you decrease, or worst case scenario remove this velocity, they begin to spiral inward, towards the Sun.
This is exactly what is proposed by the Nice model to have happened with Jupiter.
As it did so, it crashed into the rock that will eventually form the 4 inner terrestrial planets (not so good for our existence...)
But something changed; something halted what seemed like an inevitable path right through the rock that would later have formed Mercury, Venus, Earth and Mars.
That something was Saturn.
The birth of Saturn
Image credit: NASA
Saturn formed at the right place, and at the right time.
As Jupiter started to reach the region of what we now observe as Mars' orbit, Saturn became large enough such that, in its close proximity to Jupiter, its strong gravitational field was able to, over time, pull Jupiter back.
Note however, that this would've been a very gradual process over millions of years, as, due to Jupiter being closer to the Sun than Saturn is, for every time Saturn completed 2 orbits, Jupiter completed 3.
As it would catch up to Saturn, Saturn would pull at it with its gravity causing it to gain momentum in that direction, away from the Sun.
Even-though Jupiter would've been (and still is) more massive than Saturn, the two planets would've been caught in this situation where their centre of gravity, or barycentre as it's otherwise known as, would've been somewhere between them, but more levied towards Jupiter since it's the more massive object.
Therefore, Jupiter would be accelerated outwards, away from the Sun, and Saturn would be accelerated inwards, towards the Sun.
What's particularly interesting though, is that these 2 objects played a massive role in the order of the solar system today.
Their combined gravitational strength largely effected the surrounding debris' ability to clump together and form new planets, as they were constantly being tugged at by the 2 gas giants.
Additionally, every time they reached their closest point with one another, their gravitational fields would overlap, and "double up" if you like, causing evermore mayhem for the debris that would later form Neptune and Uranus.
Anyway, as Jupiter and Saturn slingshot debris everywhere, this material interacts with the 2 next planetesimals, Uranus and Neptune, such that they're pushed outward due to a conservation of momentum.
What's really interesting here, is that the model suggests that Neptune actually formed closer to the Sun than Uranus, but interactions with the combined strength of Saturn and Jupiter caused it to be pushed outwards, eventually overtaking Uranus and becoming the furthest planet from the Sun.
The fact that Neptune is slightly heavier than Uranus (about 20% more massive) aligns with the model, as debris becomes less dense the further you go from the Sun, hence you would expect the least massive objects to be the furthest away from the Sun.
Without this migration theory, we're unable to explain why Neptune orbits further out from the Sun than Uranus, despite being heavier in mass.
As this is all happening, inner rocks begin to collide and glue together to form the terrestrial planets.
What's important to realise though is that while Jupiter edged towards what now is Mars' orbit, it swallowed up a considerable amount of the material, which perfectly explains why Mars is so much smaller than Venus and Earth.
That pretty much sums everything up...
I hope it was an interesting read and feel free to stick around and check out some of my other posts!