The Brainliest Answer!
Air is present inside a balloon at a pressure higher than atmospheric pressure. 

When the balloon is pierced from the bottom, air from the balloon comes out which moves downward(as the hole is made at the bottom of balloon). So the balloon exerts a downward force on the air. According to Newton's third law, the air will exert an upward force on the balloon. Hence the balloon will rise up.

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But, my teacher is saying that it's law of conservation of momentum.How could it be law of conservation of momentum?
It can be law of conservation of momentum too. But Newton's third law suits better.
law of conservation of momentum is applied when there is no external force. but here, there is gravitational force. So theoretically, you can't apply law of conservation of momentum.
But gravitational force is negligible in this case. So you can say that there is no external force and hence the momentum is conserved i.e. initial momentum = final momentum.
initial momentum = 0 as it was at rest initially.
So final momentum will also be zero. Since the air going out has downward velocity, the balloon must have upward velocity so that the system will have 0 moentum.