How Did Jesus Break The Power Of Satan Over Humanity?

Satan caused the fall of the first Adam. He was hoping to succeed with the second Adam - the man Jesus Christ.

Satan approached him through a series of temptations while thus weak and suffering, tempting Jesus to do a miracle for his own sake.

Thus Christ engaged in a conflict with the prince of this world. He was fighting the battle on behalf of humanity and endured the most terrible temptations. 

Jesus placed Himself in the position of Adam and endured the test where Adam had failed.

Humanity would have been lost had He faltered, yielding to temptation.

Thus Jesus became the mediator for humans through the perfection of 

His character.

Accepted as such by the Father, He only could save us by imputing to us His righteousness.