Refer to figures. 
The answer may seem surprising at first, but the key point here is that in the climb, it is excess thrust that provides the upwards force, and not excess lift.

The figure shows us the forces in a steady climb. The thrust is inclined upwards - and therefore counteracts the weight slightly, allowing the total lift to be reduced. As the angle of climb increases, the lift actually reduces even further. Mathematically, we have:

L = W cos(γ)

The lift only has to balance W cos(γ). For any climb angle γ, this is less than the weight!

This can be even better understood by imagining the case of a 90° vertical climb - the thrust carries all of the aeroplane's weight, and counteracts the drag, so there is no lift being produced at all!