The 109 series was a very forward-thinking design, and the tactics they developed became the foundation of basically all modern Aerial combat. (high-speed, boom and zoom)
The Spitfire was one of the best of the "old" designs. (highly maneuverable, turn-type dogfighting.)
The Bf-109 could outrun almost any other plane of the period and had a very narrow rear profile. Also, it had a fuel injector, so the engines did not die if you dove too sharply, which was a problem for the British planes.
I disagree with some of the above. The 109 had severe limitations imposed on it by the basic design, and while ingenuity kept the plane at least serviceable, it was quite dated by the latter half of the war. I would go so far as to say that the 109's widespread deployment had a negative impact on Germany's aircraft weapon development, as it simply couldn't carry some of the more theoretically effective designs on the drawing board, and if it couldn't carry them why bother making them? (Or so the thinking went, in some circles.) Kurt Tank (and others) realized this even before WWII began, hence the FW-190...which still couldn't carry some of that stuff, but that's another story.
The German boom-and-zoom method of fighting was adopted out of necessity, as starting with the Friedrich (or was it the Emil? It's been a while) it could no longer turn with its adversaries, and the angles game ceased to be an option. "Turning and burning" was an easier way to fly and only really started to leave most nations' doctrines with the advent of jets, and the Russians never actually gave up on it. It seems to be making a comeback today.
Carburetted engines were more of a refinement than a serious advantage at the start of the war. True, a hard enough dive would cause the engine to sputter or even stall completely, but hard negative gees are also rather unpleasant (and even dangerous) for the pilots. A quick half-roll to inverted made the dive a climb as far as the fuel injection system was concerned and pilots were trained appropriately. It was something that could easily be "designed in" later on down the development line when convenient.
Basically, if you can trick a german pilot into a turning dogfight, you might win in a spitfire. If you don't see him coming and you don't have enough altitude, though, you are dead meat.
They sometimes did not have the choice.
Historically in air warfare, not seeing the enemy coming usually got one shot up regardless of one's altitude, or what aircraft they were flying. The aces that lived to write about it, generally tell of surprising their enemies and attacking from positions of extreme advantage when at all possible. (Which is probably why they lived to tell about it.)
Remember, Chuck Yeager bagged his 262 while it was trying to land.
As to which one was most successful - The Israeli Air Force, when it was created in the late 40's, used the BF-109G or K, I think, which was one of the last models of that plane with a pressurized cabin for hi-altitude flight.
Only until they were able to get Mustangs and Spitfires.

There was probably a cultural aspect involved in this decision though, and Israel's S-199's are often mentioned anecdotally as dogs at the best of times, and widowmakers at the worst. They are not fair representations of wartime 109's.