If you define win as who achieved their aims, then I can't see how the US won the War of 1812. In order to understand the war you have to understand that in 1812 the Napoleonic wars were raging. The British had pretty much total control of the sea but British trade was subjected to Napoleon's Continental system- a land based economic blockade of the Britain. Their armies were tied down in Spain and Portugal fighting the Peninsular War- a sideshow of the Napoleonic War really. Napoleon's Grand Armee invaded Russia in 1812. In essence, in 1812, the war was not going well for Britain. Prussia was still an ally of France and the Sixth Coalition, the alliance that ultimately defeated Napoleon, had not yet been formed. The British public was not very happy about the war. 1812 was an election year and also saw the assassination of the Prime Minister, Spencer Perceval. Incidentally, he was the only PM assassinated in British history although his assassination was not related to the war per se. In an effort to stem the tide of British sailors jumping ships, the British Navy adopted a policy of stopping any merchant ships in an effort to find deserters. Needless to say, the American's were somewhat unhappy with this and this as an excuse to get back at England.
British North America was almost undefended at the time given the events elsewhere in the world. To the US, 1812 seemed like the perfect time to gobble up the top half of the continent through "a mere matter of marching". So, there goal was to conquer British North America.
The British government's goal was to defend Canada in total if possible but if not, to at least hold on to Canada from Quebec City eastward.
In essence, it is hard to see how this war could be considered a win for the US. When faced with overwhelming odds on paper, the small British garrison, augmented by Native allies and some militia, managed to take advantage of what would be almost comical, if they weren't so tragic, invasion attempts by the US until British reinforcements arrived later in the war. At the close of the war the British forces had capture Detroit, parts of upstate New York and burned the White House while sacking Washington, DC. Their only partly successful invasion attempt occurred in the Niagara campaign and the raiding of York but these were quickly repulsed.