homework and exercises - Why doesn't the Gravitational Force causes body to fall towards orbital center moving with constant tangential speed?

Suppose a satellite is orbiting the Earth. The gravitational and centripetal force supposedly point towards the Earth. Therefore, the net force is towards the Earth. Since the satellite doesn't fall immediately towards the earth, what's the third force holding the satellite in orbit? First impression tells me its the elusive "centrifugal force", but I was told to use this with caution as it's often a misnomer. What's the correct line of reasoning behind this?