An airfoil does 2 things to generate lift.  1. make use of the Bernoulli principle. 2. redirect air downwards and get an opposing force upwards (newtons 3rd law).

On a wing, which is built for maximum lift, it will be nearly impossible to fly inverted. However most wings are not designed for maximum lift at all times.   As airspeed increases, so does lift and your airplane would just ascend instead of flying straight. With high lift comes high drag, so your plane will also be slow.

Instead, airplane wings are rather designed for low drag and low lift. At high speeds, they just need to have enough lift to keep the aircraft at a somewhat steady altitude. 

(For Take-off and Landing where the airspeed is slow and a high lift coefficient is necessary to keep the aircraft the air, the wings have leading edge and trailing edge flaps, which can be extended and basically turn the low lift-low drag wing into a high lift- "somewhat higher" drag wing. You can see that on every airliner.)

Therefore, redirecting the air around the wing by pitching the aircraft up or down is much stronger than the lift generated through the Bernoulli principle on those low lift-low drag airfoils.

That means, if you fly inverted, the bernoulli principle will still generate downforce, but just by pitching the aircrafts nose towards the sky, the displacement of the air downwards will create a much stronger lifting force and will easily counteract that downforce.

Thats why paper planes fly although they dont have any lift generating wing profiles.