I haven't come across a system I'm particularly happy with. I'd like to see a system that captures the ebb and flow feel you get from fights in cinema or from what little boxing I've seen. One side has the advantage and presses it until they try to land a telling blow, and either make it or fluff it and let the other guy come back. There's tension in the choice between playing it safe and gradually wearing the other guy down, or taking a gamble to put him down quickly.
Ars Magica 4th edition does something like that. It still uses conventional initiative (d10 + modifiers, but modifiers are large enough that the d10 can become little more than a formality), but both to-hit and damage are opposed rolls (Attack vs. Defense and Damage vs. Soak, respectively), with the margin of success on the to-hit roll being applied as a bonus on the damage roll - if you choose to do damage. Instead of doing damage, you also have the option of carrying that margin over to the next round and using it as a bonus on either your Attack roll or your Defense roll, thus allowing you to build up an arbitrarily large carryover bonus over the course of multiple turns until you finally blow it all to land a solid blow.
I've also read (but not played) another system which does what you describe - unfortunately, I can't seem to recall what system it was - by implementing a split similar to the split between "luck HP" and "meat HP" that some have implemented as house rules in D&D. The "meat HP" are a static quantity for each character, but the "luck HP" are rolled anew at the beginning of each fight. In order to inflict meaningful damage, your damage roll has to exceed your opponent's remaining "luck HP", but attacking "luck HP" is a different action than attacking "meat HP" and, when after you attempt to damage the "meat HP", they reroll their "luck HP", regardless of whether you do actual damage or not. So the flow of combat would be that you repeatedly attack their "luck HP" to wear it down until you think you can successfully get through to inflict actual damage on their "meat HP", then the "luck HP" resets and you fence for position (wearing down the "luck HP") again. The "luck HP" in this system is called Stance, or Poise, or something like that, making it clear that the process this is intended to model is that you're working to create an opening in their defense and then, when you attempt to exploit that opening, they reposition themselves, and cover the weakness you targeted.
But, then, neither of those subsystems are based on initiative, and neither game is OSR (in the "emulating early-era D&D" sense)...