Acts 2:39 For the promise is to you, and your children, and to all who are afar off...
A lot of our argument comes from the inference that Baptism is to Circumcision as the Lord's Supper is to Passover.
Colossians 2:11-12 In [Christ] also you were circumcised with a circumcision made without hands, by putting off the body of the flesh*, by the circumcision of Christ, having been buried with him in baptism... (emphasis mine)
Just as circumcision was the "putting off the body of the flesh", so baptism is being "buried with Christ", "into death" (Romans 6:4). They are signs of the same promise of the same salvation, one given to the physical race Jews, the other to the world-wide race of the elect.
Since the rite of circumcision was performed on male infants at eight days old, it makes sense to retain the age for Baptism.
Genesis 17:12 For the generations to come every male among you who is eight days old must be circumcised, including those born in your household or bought with money from a foreigner—those who are not your offspring.
Note that those who are baptized are those "in your household" or those brought into the household, so by carryover this would only apply to children of believers in Baptism.
Just like we (usually) retain the general "age of accountability" for Lord's Supper (just like Hebrew boys generally were supposed to wait until they were 12 to take Passover), so it makes sense to keep the age for Baptism.
While the following verses do not stand alone to support infant Baptism, they support it because they do not reverse the age set in the Old Testament for circumcision.
Acts 16:14-15 ...The Lord opened her heart to pay attention to what was said by Paul. And after she was baptized, and her household as well...
and
Acts 16:33 And he took them the same hour of the night and washed their wounds; and he was baptized at once, he and all his family.
While you would be right to say "ah, but this is an argument from silence," I would reply that it certainly stresses "her household" and "his family" and considering the emphasis given in the New Testament to making sure that the Jews understood that the sacrifices/ceremonial law were now nullified, it would seem foolish of Luke not to carefully point out that the infants and children were omitted in the baptism. Instead, he includes the household and the family in the event because the parents (particularly the father) were representative of their family.
When God makes his covenant with Abram in Genesis 17, he says "offspring" five times before giving the command to circumcise all males, beginning at those only eight days old.
1 Corinthians 7:12-16 suggests this has not been nullified for the New Testament and foreward; vs. 14 says
For the unbelieving husband is made holy because of his wife, and the unbelieving wife is made holy because of her husband. Otherwise your children would be unclean, but as it is, they are holy.
My pastor, in a letter to our congregation, said:
Children of believers, like adult believers, are members of the visible church (church militant); and they, like we, are members of the invisible church (church triumphant) only by grace, through faith.
He has also called Baptism an engagement ring, placed upon the finger of the children of the covenant, and awaiting acceptance or rejection by that child as they grow.