Yes, the Sacrament of Confession is biblical — as all the other Sacraments are.
There are, of course, many things that Jesus said and did which were not written in the Bible. As St John said to end his Gospel: “…there are also many other things that Jesus did; if every one of them were written down, I suppose that the world itself could not contain the books that would be written” (Jn 21:25).
Many of these unrecorded stuff, my friends, are kept carefully by our Mother the Church throughout the centuries. These form part of what we now call Sacred Tradition.
BUT, the Sacrament of Penance or Confession certainly is among those things Jesus explicitly said!
Before ascending to heaven, he commanded his disciples: “As the Father has sent me, so I send you. […] If you forgive the sins of any, they are forgiven them; if you retain the sins of any, they are retained.”
And in terms of Confession being the reconciliation of the sinner with the Church, the Catechism reminds us of what Jesus said when he instituted the primacy of St Peter as ‘prince of the apostles’: “I will give you the keys of the kingdom of heaven, and whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven” (Mt 16:19).
The Church has always understood this “loosing” as excluding the unrepentant sinner from communion with the Church, and “binding” as restoring through Confession the contrite one to communion (CCC 1445). Of course, this “binding and loosing” power cascades to all bishops and priests united to Peter — that is, the Pope.
So, yes, even if Jesus never said that we should believe only on what the Bible says, Confession is biblical and is indispensable.