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In many sci-fi or sci-fantasy works (Star Wars, Babylon 5, Earth: Final Conflict to name a few), space travelers go to far off systems by moving through some other dimension, reducing a trip that in 'real space' would take centuries, if not millennia, to a much shorter time- maybe even days or weeks- what was the first example of this method of space travel in fiction?

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    E.E."Doc" Smith did this is several book series. Lensman (1937), Skylark (1920) IIRC
    – Paulie_D
    Commented Aug 11 at 18:51
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    @Paulie_D Are you sure? Its been quite a while since I read them, but I recall them traveling through "normal space", just ignoring the speed limit.
    – user14111
    Commented Aug 11 at 20:14
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    There were kind of "boom tubes" used in certain instances and Skylark had the protagonists "rotating through other dimensions".
    – Paulie_D
    Commented Aug 11 at 20:29
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    Would John Carter's magical trip to Barsoom be considered an interdimensional shortcut? "A Princess of Mars" was published in 1912. Commented Aug 11 at 21:23
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    @StanleyWebb not disputing the trip is magical, I'd call it "unexplained teleportation". Commented Aug 12 at 10:11

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The earliest mention I can find of something like this is in John W. Campbell, Jr.'s Islands of Space (1931):

"Well, we would be stuck if we didn’t have this new method, for the acceleration would take too long. However, in the hyperspace we are going into, a new condition exists. In this space, you never saw light standing still, did you? Now, when light is made, it is at rest, and yet it instantly attains a velocity of 186,000 miles a second. How? Light cannot go slower than that speed and exist. However, the presence of certain things in space will alter the minimum velocity. For instance, it can go more slowly through glass, for the presence of the glass alters the condition of space. Well, in this hyperspace we are creating, matter cannot exist at a velocity lower than a certain quantity, and we determine that quantity by using this apparatus. We are subject to an infinite acceleration—and we are subject to that acceleration in every particle. Remember, that we have no weight when we offer no resistance to gravity, but we yield and fall freely. We yield to the infinite acceleration, and are not hurt by it. Result—we move at the velocity we desire," explained Arcot.

"Well—you go into a hyperspace—and move at any speed you please. I wonder then, how are you going to see where you are going?"

"We won't. While in that hyperspace I don't expect to see anything. We will start the ship in the direction we wish to go, then go in that line when we fall into hyperspace, and since we direct our path, we can avoid stars in that way. Meteors, or objects too small to be seen, we need not worry about, for the old law correctly stated, is, two bodies cannot be in the same space at the same time. We won't be—we will be in a different space.

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