I think Putin's wrong if he thinks that. The ideal for Russia is probably to have stability along the European border as well. As long as the Near Abroad is scared of the bear they are going to gravitate to its adversaries. But in eastern Europe in particular there's a ton of resentment against the west, and if Putin would just cool his jets (literally!) a casual, comfortable environment would give him the trading partners and commercial markets necessary for him to keep the Russian economy going -- the only thing that will keep the population from looking to a different strongman.
On the other hand, I'm surprised Putin hasn't been playing games in the Caspian against the Iranians. He can ratchet up a Forever War of his own to justify authoritarian rule (and where have we seen that trick before?), sell arms to everybody in the region, and use the unrest to keep Ma and Pa чайник distracted from the high cost of vodka.
