So, I have been examining this issue both in KSP, and from the mathematics behind how rockets perform.
Simply put, in all cases, staging (specifically "asparagus" staging) is the best way to get the most Delta-V out of any rocket.
If you aren't familiar, check out the simple formula for The Rocket Equation
This equation describes how much Delta-V a given rocket has. So let's take a closer look...
Dv = Ve * Ln (Mt / Me)
Dv = The total Delta-V
Ve = propellant velocity
Ln = Natural Log function
Mt = total mass, Including propellant
Me = empty mass, without propellant
Starting with Mt / Me, it's important to note that although this part of the equation is measured in Mass, the units end up falling off and you end up with a simple ratio; the full to empty ratio. What this means is that it doesn't matter how big or small your rocket is, when calculating Delta-V, only how much of it is made up of Propellant.
The Ln function gives us a diminishing return for improved ratios.
Ve is the effective exhaust velocity of the propellant, often measured in meters per second, but most rocket engines list an Isp (specific impulse). Since Ve = Isp * G it's easy to plug this directly into the equation.
With either Isp or Ve, this does not describe the force put out by the engine in any way - just the efficiency of that engine - so a more powerful engine (given equal mass) does nothing to change your Delta-V. In reality a more powerful engine has more mass, which actually reduces the total Delta-V of the system.
Now... all this context is fine and all, but what about your question? Does using Staging actually improve Delta-V?
Yes, and here is why:
A single stage rocket uses the Rocket Equation:
Dv = Ve * Ln (Mt / Me)
The same is true for any particular stage in a multi-stage rocket; that stage uses the same rocket equation:
Dv1 = Ve * Ln (Mt / Me)
Dv1 = Dv for stage 1.
Since any mission is the work of multiple stages, you can simply add the stages together to get the total Delta-V for the mission:
Given N Stages:
Dvt = Dv1 + Dv2 + ... + Dvn
So - armed with the math, let's do some calculations!
Suppose you have a single stage rocket that is 80% propellant, with an effective exhaust velocity of 3234m/s (a Rockomax Mainsail in Vacuum). Remember, the actual mass doesn't matter - all we need is the ratio... But we'll plug in the masses to get there so our brains can keep up.
Dv = 3234m/s * Ln(5tonsFull / 1tonEmpty)
Dv = 3234m/s * Ln(5)
Dv = 3234m/s * ~1.61
Dv = ~5207m/s
Now, let's break this up into STAGES. Remember Dvt = Dv1 + Dv2
First Stage uses 50% (2 tons) of the available propellant:
Dv1 = 3234m/s * Ln(5tons / 3tons)
Dv1 = ~1652m/s
Second stage drops off 0.5 tons empty mass during decoupling:
Dv2 = 3234m/s * Ln(2.5tons / 0.5tons)
Dv2 = 3234m/s * Ln(5) (Doesn't this look familiar?)
Dv2 = ~5207m/s
Now total the stages:
Dvt = ~1652m/s + ~5207m/s
Dvt = ~6859m/s
Now, we can keep doing so for as many stages as we want (within the limits of the technology we have). Every time we divide the rocket into stages, we increase its delta-v. Asparagus staging is an expression of this mathematical principal.