George Hunsinger, in his really superb book How To Read Karl Barth: The Shape of His Theology, sketches the difference between Augustinian (classical) salvation, and what Barth offers as a better (in my view) 
puzzlealternative. In this sketch what emerges is very profound, and should help to crystallize further what Myk Habets and myself (and I have been frequently on my blog here over the years) have been trying to articulate in regard to what we have been calling a 'Christ-conditioned' or (pace Barth) 'Christ-concentrated' approach to a doctrine of God, and then subsequent things like in regard to salvation in particular. What emerges in this sketch — at the end of it — is the stumbling block point for so many of the people I have engaged with around this particular question over the years; and usually it stumbles people because they are still trying to appreciate evangelical Calvinism, they are still trying to appreciate Barth through Augustinian and classical categories of conception — I hope what I am about to share from Hunsinger on Barth will finally help to make clear why trying to read us (the Evangelical Calvinists and Barth) this way simply will not work, and in the end always terminates in frustration and not fruit. So without further ado, here is Hunsinger on Barth and the difference in approach between Augustinian trajectory and Barthian trajectory:

Three things are to be noted about the universalist direction evident in Barth's objectivist soteriology. First, the salient difference between Barth's position and more traditional views has primarily to do with the locus of mystery. The great puzzle for more traditional views like those of Augustine or Calvin is why God should decide to save some but not others. Regardless of where the decision is thought to be taken—whether in predestination, in the cross, in the convergence of grace and faith in the individual's spiritual life, in the last judgment, or in some combination of these and similar factors—it is still understood to be primarily a mystery about the inscrutable good pleasure of a deity who loves but does not save all (or who condemns all yet still saves a few). By contrast, the great puzzle for the Barthian view is more nearly anthropological than theological in location. The divine disposition, decision, and work for our salvation are presented in unequivocal terms. No disposition, decision, and work of God are to be found elsewhere than in Jesus Christ, who died to cancel our past, rose again to establish our future, and pleads for us to all eternity.

The mystery pertaining to God as such is not the puzzle of an inscrutable decision to save some, but not all. It is the mystery of an unfathomable mercy that (at great cost) saves all, not just some. But there is still a puzzle of what might be called the "dark mystery," and it corresponds to the puzzle embedded in the more traditional views. The dark mystery for the traditional view is, as noted, that God does not will to save all. For the Barthian view, however, it is rather that not all human beings will to accept God's salvation. The dark mystery is that human beings inexplicably (i.e., "inexplicably" within the terms of Barth's tlnelogy [sic]) are all by appearances actually capable of rejecting the divine disposition, decision, and work in their favor. It is the puzzle of our rejection of grace, the mystery of sin, but here raised to a very high pitch, since salvation is somehow effectively rejected even though it fully avails for those who reject it. This is not the place to explore the intricacies of Barth's conception of sin as a dark mystery. The point is simply that the problem of an inexplicable puzzle has been shifted but not eliminated. The puzzle for the more tradition view is that God's will seems to be truly inconsistent. For the Barthian view it is that the human will to reject the divine grace, while actual, would appear to be truly impossible. It is Barth's contention that the gospel finally leaves us with just this mystery, and not with some other "very different mystery" (IV/2, 520). [1] George Hunsigner, How To Read Karl Barth: The Shape of His Theology, 129-30.

I often come across things like this, and I am reading them I think: "Sweet, this is awesome, if I share this with folks, this ought to clear an awful lot up in regard to where I am coming from in terms of my Evangelical Calvinism." Then to my chagrin, usually the response is opposite. I would have come to think it is because of what I already mentioned in the prologue of this post; viz. that folks are trying to think scholastically and not dialectically about such things. In other words, folks are still so trained and conditioned by their Augustinian and classical way of thinking about these kinds of questions, that trying to negotiate with what Barth is saying, and what we Evangelical Calvinists are saying in these regards, again, becomes a point for frustration and often times, caricature, instead of an actual appreciation for what Barth and us Evangelical Calvinists, at this point, are trying to communicate — and so the result is that we keep skipping off of each other, and never are able to enter into the same door, to inhabit the same space, wherein actual table-talk and Christian fellowship can prevail (which does not finally mean ultimate agreement about such things). Further, I think another thing that hinders (and it probably should at some level), is that we have inherited the classical/traditional Augustinian approach to a doctrine of salvation so uncritically through our churches, bible studies, and hymns (and choruses); that any other or alternative ideas that might be offered to what we are used to, are automatically considered non-Biblical, and even against the gospel itself.

What I would like to ask of us, is that we would really consider the conceptual stuff itself; on its own merits. This of course will require way more than just reading this blog post (or even multiple ones). You owe it to yourself to genuinely and critically consider these things. If after that you still can affirm the more Augustinian tradition course in regard to thinking about this particular question and salvation in general; then of course, by all means, continue moving forward in that direction. But I have been attempting to (over the years) do this (i.e. critically consider these alternatives), and what I have concluded is that Karl Barth, Thomas Torrance et al. really do offer a better more Christ concentrated way forward relative to the question under consideration in this post, and then of course, in general, in regard to a doctrine of God (which implicates everything else that follows!).
I hope this post has clarified some things. If it hasn't, then please tell me why it hasn't. The point of departure among us, in the end, will be an issue of heremenutical variance and prior theological commitments (which is what I just said above). Just bare this in mind.