Captain America is a Marvel character. As well as the groups mentioned by Postal, Marvel (strictly "Timely" - the company kept changing its name) comics of WW2 also set the Sub-Mariner, the original Human Torch, the Destroyer and others against the Nazis. Since WW2, the main other Marvel comic set in the period has been Sgt Fury and his Howling Commandos.
DC came up with the "Spear of Destiny" thing in the 1970s to explain why Superman hadn't ended the War in an afternoon. I think there was an equivalent Japanese artefact too. But back in the 1940s, their superheroes did sometimes fight the Axis: Wonder Woman, for example, came up against Japanese in the Pacific and U-boats (and, of course, the first run of the TV series was set in WW2). In fact, there was actually a 2-page Superman strip in Look magazine which he did end WW2 in an afternoon by arresting both Hitler and Stalin and putting them on trial before the League of Nations. Innocent days!
Roy Thomas, who created The Invaders for Marvel in the 1970s, drawing together all their WW2-era superheroes, did the same for DC in the 1980s with All-Star Squadron and the Young All-Stars. DC also once had a parallel world called Earth-X where the Nazis had won WW2 and were being opposed by a group of super-heroes called the Freedom Fighters, whose rights DC had acquired from Quality Comics.
Over the years, DC has also published many originally non-Superhero WW2-set comics, which have now been more or less integrated into the regular DC Universe. As well as Sgt Rock, look out for The Losers, Mlle Marie, The Haunted Tank, Hellman's Hellcats and Weird War Stories.
American comic books are preposterously well-documented on Wikipedia, so you'll find loads of info - and lots of links on to other characters - there by looking up any of these names. Note that some of the names have been reused. The Losers has been recycled as a modern-day spy story, and The Haunted Tank has a sequal set in Iraq, for example.