Children are saved in the same manner as all other people: through faith in the finished work of Christ to atone for their sins.
I believe I know where you want to go next. What about when an infant (let's go even further and say child in the womb) dies? The only Biblical answer to that is found in Romans 9:19-24: "You will say to me then, “Why does he still find fault? For who can resist his will?” 20 But who are you, O man, to answer back to God? Will what is molded say to its molder, “Why have you made me like this?” 21 Has the potter no right over the clay, to make out of the same lump done vessel for honorable use and another for dishonorable use? 22 What if God, desiring to show his wrath and to make known his power, has endured with much patience vessels of wrath prepared for destruction, 23 in order to make known the riches of his glory for vessels of mercy, which he has prepared beforehand for glory— 24 even us whom he has called, not from the Jews only but also from the Gentiles?"