This is a common misconception. When you apply a force upward on the object, the "reaction" force in Newton's 3rd Law is NOT the force of gravity down on the object; they do not have to be equal, and as you said, cannot be equal if you are to accelerate the object upwards.
It is just a confusing coincidence that the force of gravity kind of looks like a reaction force, but it's not. One key giveaway that you have not identified a "action/reaction pair" is that both of the forces are on the same object - the force of your hand on the object, and the force of gravity on the object. Those can't possibly be a "3rd law pair".
The more accurate way of saying Newton's 3rd Law is this:
"If object A puts a force on object B, object B puts an equal and opposite force on object A, and the forces are the same type, and occur at the same time".
In your situation, those two forces are part of two separate action/reaction pairs;
- the reaction force to you pushing up on the object is the object pushing down on you
- the force of gravity on the object (from the Earth) is a reaction force to the force of gravity on the Earth from the object.
You can put an arbitrarily large amount of force on the object, and the force of gravity opposing you on the object will remain the same. On the other hand, the reaction force from the object pressing on you will become equally arbitrarily large. Luckily, you plus whatever you're standing on (probably the Earth) is way more massive than the object is, so the reaction force produces a negligible acceleration on you & the Earth, but you produce a large acceleration for the object (F = m a).