They could see the rage in Nebuchadnezzar's face, but they also saw God. Where did their powerful conviction come? This kind of conviction does not arise "on command," at the spur of the moment. It is the product of the demonstration of God in the lives of these three young men before this time, before their lives were on the line. Their faith had grown and matured over a period of time.
God is always the same. What God says through Paul in I Corinthians 10:13 applied to them just as it applies to us. God knew what they could endure. They also knew that He would provide a "way of escape." Because of this, they told the king, despite his threats, "Even if God does not choose to protect us, we still are not going to bow down to your image."
Have we ever considered why more "mighty deliverances" do not occur to us? It is because we spend so little time fellowshipping with God that we do not see Him as an immediate and vitally important part of our lives. As a result, the physical "evidence" we see around us overwhelms us.