If you take a close look at the final outcome it no way can be said that it was an American victory considering none of their war aims were achieved.
Besides, if your conclusion of victory is based on the result of the "final chapter" of the war then the American disaster and ultimate surrender at Fort Bowyer after being attacked by a much smaller British force three weeks after New Orleans (these British troops had fought at New Orleans too) then surely, by your own reckoning, the laurels rest with the anglo-canadians.
However, there is no mistaking that the war came close to becoming a major victory for the anglo-canadians. The peace terms imposed on the Americans could have been very harsh considering at the end of the war British and Canadian forces held large parts of north east America; the Royal Navy had arrived in massive force and could block any port in America and land large armies anywhere they pleased along the eastern seaboard. While the Americans had an excellent navy with superbly made ships, crewed by very competent men, it was tiny compared to the British one. For example, the largest American ships had 40-50 guns whereas the largest British ships would have 90-100, and they had over a hundred of these monsters plus a further 500 ships of other various sizes. The Royal Navy would remain patrolling US coastal waters for the next 100 years until the outbreak of WWI.
The fact that the Americans were being stopped from trading with Europe (mainly France) by the British was one of the main causes of the war. However, the Americans in the North East were still eagerly trading with Britain and really did not want any war. In an ironic twist of fate the "Rebels" of the Revolution turned into the "Loyalists" of 1812. During the war the towns along the New England coast still maintained a steady trade with Britain, even selling them timber for their ships and supplies for their crews, Even after the Royal Navy arrived and blockaded the ports English ships were still allowed though and continued to trade The southern states were more pro-war due in part to the fact they had the new recently purchased non-English states/colonies in their ranks. Louisiana being a prime example, where the Battle of New Orleans took place - the main reason why many Americans believe the war was an ultimate victory for the US; despite it taking place after the peace had been signed and as per the revolution much of the American Militia being made up of former French colonists.
Thankfully for all concerned the British had just finished an 18 year war with the French and were more concerned with getting out of America and using the hard won peace to expand their Empire and the former American colonies cost them more money than they could ever recoup, even when the colonists behaved themselves. Plus, with the defeat of France the need to stop anyone trading with europe was removed, as such the peace terms were very generous to the US.
All sides finally sat down and agreed a border between the USA and Canada, which still remains to this day; and as a thank you for the support of the people in the ports of the north eastern seaboard the British allowed them fishing rights in the St Lawrence river, which was a great help as at the time since fishing stocks off the coast of New England were shrinking.
The main thing for America was the fact that this was a lesson learnt. Because of the confidence gained from the War of Independence, the US had been thinking it was strong enough to deal as equals with the European superpowers of the time and that it could rely on France or Spain to help them. The War was a serious wake up call - they could not take on the British alone without several other European countries willing or able to help them. Consequently America remained silent and kept out of British view for for almost a century.