I had quite some problems with this scenario, though guessing by the relative popularity of this thread, I must've been one of the few.
Still, some hints for those who might find them useful.
1. Starting from turn 1, move a unit towards the western supply dump. Force a path with your tanks, and you'll be able to capture it by turn 4 (2 turns to get to the river, 1 to cross, 1 to capture). I used my cavalry, but a carefully guided infantry unit might work too.
2. Protect your armour in the northwest with your infantry - the AI really likes killin ur d00dz when given the chance, and this allows your own sickly tanks to survive a bit longer. You can't hit the SS Panzer Corps head-on anyway. Destroying the Grossdeutschland division on the first turn is optional, but I prefer to encircle Poltava (only Poltava, not everything north of the bridge) and keep them and the SS unit OoS.
3. The Panzer Corps' movements can be a bit erratic, but try to cut them off whenever given the chance. Destroy any infantry in your path instead. You should be able to get to the bridge on turn 4. In my last game, the SS units, for some reason, moved toward the east to cut off two infantry divisions heading directly towards Dnepropetrovsk. If they instead gather around the bridge or even cross the river, I'd say you're screwed unless you've cut off both enemy supply dumps.
4. Take the bridge and move what you can towards the east.
5. Around Stalino, things can get a bit hectic. I've tried several times to go for the grand encirclement, only to find my spearheads cut off or blocked by the massed Axis armour and mobile units. Especially the elite Wiking division is a tough nut to crack, and it's not like you can quickly move a large number of units west to support your tanks. Instead, I went for a 'little solution', where I hugged the town of Stalino with as many units as possible.
6. The enemy will now put its heavy hitters south of yours, but your relative position allows you to cut them off again further to the west. By the next turn, all those units are hopelessly OoS, so take out anything that might still pose a threat if it escapes, and start moving west. Force your Mech corps through the cauldron to support your other armour, and destroy anything still standing with your infantry.
7. Don't get bogged down in the mud, so position your units carefully. I don't really understand how Thales managed to win on turn 6 (or hints at turn 5) without outrunning his supply lines (cycle-in cycle-out is a great tactic, but if the mud advances one hex too far, your spearhead is suddenly as useful as a bottle of wine on a beer festival), but all the better for him, of course.
8. Dnepropetrovsk. You've got that handy little infantry unit doing not very much at all in the centre of the map. Keep it there at the edge of your supply lines, and use it to take Zaporozhye when the opportunity arises. The AI doesn't protect the city, and it's your key to starving and taking the final target. Use whatever units you have at your disposal to get to it.
And there you have it: brilliant victory, no prestige spent. I've been playing this game a bit too much, I think.