Finey was the first to really make use of the altar call. Moody adopted his practice from Finey.
In any case, there is no real biblical precedence for the practice, but that does not make it wrong -- we do a lot of things in our local congregations that do not have biblical precedence. Even the churches that major on only doing what the Bible says do so -- like meet in a dedicated building and sit on pews, for instance. But, that isn't the OP question, I digress.
I have run the gamut in altar call, from a very Arminian, high pressure, "you are going to die and go to hell today if you do not respond RIGHT NOW" sort of approach to a very minimal approach where I simply ask people to respond to whatever God is asking them to do, either here and now (altar is open to the people) or to see me or one of the other members. Questions, counsel, adequate time to consider, prayer, etc., are all part of the process, and if one holds (as I do) that God is doing the work of salvation, then there is no need to convince, cajole, etc. He will lead as He sees fit. Our job is simply to receive.
In our current congregation (Sojourn) the altar call is made during the Communion phase of the service. We practice the Lord's Table at every service and we carefully explain that the Table is for those who are in Christ, but if one is not in Christ, or not sure of his or her relationship, now would be a good time to meet with a pastor, elder, counselor, etc., in order to deal with that issue while the "church" takes communion. At first, this practice seemed very different from that practiced in a typical Baptist church, but it grew on me and it will be my preferred method from here forward. It has certainly not hindered the growth of our congregation, we are adding 100 - 200 persons a year or more (growth is accelerating of late with more campuses more services, etc. In 10 years we have grown from 26 to 2600!).
Edit... I will walk out of the next church where I hear the pastor use the "eyes closed, raise a hand" deal, then coerces those people to walk forward to take his hand. That is youth camp crud. Christians are SUPPOSED to be brothers and sisters who lift up each other in love, and take care of, yes, even the unbelievers. How can we do that if everything is a secret (and everyone peeks anyway).