Start from the ideal gas law: PV = NkT where V is volume, N is number, k is Boltzmann's constant, T is temperature, and P is pressure. Nowhere does the mass density or mass of each atom appear.
If the Helium in the balloon is at the same temperature and roughly the same number density as the air outside, it will be at the same pressure as the air outside. In practice, the Helium will be slightly overpressured because the pressure of the Helium must balance the pressure of the air outside + the surface tension of the balloon.
Although both the air and the Helium will have roughly the same number density and pressure, the Helium will be less mass dense because each Helium atom is lighter.
The more dense air falls down around the less dense balloon, pushing it up.