This is actually quite difficult - OTL the war was launched by the Russians to take parts of the Ottoman Empire, and the British and French were coming to defend those parts of the Empire but the Ottomans beat the Russians back before the Anglo-French even arrived!
The first question is to define what "win" means. For example, forcing the Allies off the Crimean Peninsula might count.
Given that, though, the second question is how the Russians overcome a quite considerable force disparity. The British and French have rifles, and know how to use them; the Russians mostly have smoothbores with Nessler rounds, which give them a longer range than most smoothbores of Napoleon's day but not a range comparable to a rifle. Worse, the Allied fleet is so overwhelming that the Russians don't even bother trying to fight - they scuttle instead of try battling the Allies - and the Allies also deploy the world's first true ironclads. (Sail-equipped, steam powered, armoured vessels capable of independent cruising and armed with powerful shell or ball guns.)
On land, the Russians face the problem that their army is opposed by modern, up-to-date, professional forces capable of killing them from easily twice their range and of launching uphill bayonet charges against fortifications - and fully able to supply themselves even over the beach at a 3,000 mile range from their ports.
So I think it would be easiest to use a pre-war PoD.
Here's a few possible options:
a) The Russians are the ones to invent Martin's Shell, an OTL Royal Navy invention, which means that a fort can demolish (set ablaze) wooden ships very efficiently. This renders Sevastopol much more defensible, and the same is also true of Kinburn and Bomarsund and other forts.
b) The Russians purchase very large numbers of Dreyse breechloader rifles, and follow this up with proper rifle training. Even if they still stand in lines and blaze away, this will make them much more effective than the OTL.
c) Heavy Russian use of rifled artillery, outranging the British smoothbore artillery and to some extent their rifled long arms.