The question required that we reference instances of time travel that are not simply "a misperception of the amount of time that passed". One could argue that King Raivata's story in the Mahabharata was such a misperception. Others might argue differently, but here's an alternative answer for the first group:
Although not 2000 years old, there are examples from before The Time Machine (1895) that involve actually jumping from one time to another.
A Connecticut Yankee in King Arthur's Court (1889) by Mark Twain
This is the first one I thought of. Partial summary from Wikipedia:
In the book, a Yankee engineer from Connecticut is accidentally transported back in time to the court of King Arthur, where he fools the inhabitants of that time into thinking he is a magician—and soon uses his knowledge of modern technology to become a "magician" in earnest, stunning the English of the Early Middle Ages with such feats as demolitions, fireworks and the shoring up of a holy well.
This is very clearly time travel. The Yankee was in modern times, then he was in the past.
"The Chronic Argonauts" (1888) by H.G. Wells
This is a short story precursor to the Time Machine. Partial summary from Wikipedia:
A third-person narrator describes the arrival of a mysterious inventor to the peaceful Welsh town of Llyddwdd. Dr. Moses Nebogipfel takes up residence in a house neglected after the deaths of its former inhabitants. The simple rural folk become apprehensive about Nebogipfel's activities in the house and suspect him of witchcraft. ... reveals that Nebogipfel is an "Anachronic Man" whose genius drives him to seek out a time more suited to his abilities.
I haven't read the story myself, but according to @Hypnosifl, the story describes the travel as "Locomotion along lines of duration". This might be some kind of continuous movement, but considering how it's going backwards it would be hard to characterize it as a misperception of the passage of time.
"The Clock that Went Backward" (1881) by Edward Page Mitchell
This is a short story that Wikipedia claims is "the first instance of using a time machine for time travel, and the first instance of a temporal paradox in fiction". The citation for this claim links to a book by Paul J. Nahin called Time Machines: Time Travel in Physics, Metaphysics, and Science Fiction which contains a review of time travel in fiction. It describes the time travel device like so:
The mechanism of Mitchell's time machine, an eight-foot-high, sixteenth-century Dutch clock, is quite simplistic, even bordering on fantasy. It is simply stated that if the clock runs backward, then it travels backward in time - a rather disappointing explanation.
Like "The Chronic Argonauts", this appears to be a continuous motion backwards through time. According to @Hypnosifl, the story states that
The hands were whirling around the dial from right to left with inconceivable rapidity. In this whirl we ourselves seemed to be borne along. Eternities seemed to contract into minutes while lifetimes were thrown off at every tick.
Again, though, this is movement backwards. As this is movement through time in a different direction, rather than simply at a different speed, it seems to be a clear example of time travel.
That may be the earliest use of a device to travel in time, but in a Wikipedia article called Time Travel, there is a section on the history of the time travel concept, in which there are multiple possible examples earlier than 1881. The earliest of those examples which looks to me to be clearly not a dream or vision, and not someone simply sleeping into a later time Rip van Winkle style, is from 1836.
In The Forebears of Kalimeros: Alexander, son of Philip of Macedon (1836), by Alexander Veltman
the narrator rides to ancient Greece on a hippogriff, meets Aristotle, and goes on a voyage with Alexander the Great before returning to the 19th century (Wikipedia)
In my research, I ran across numerous blogs referring to this as the first Russian science fiction novel, and as the first novel using time travel. These aren't exactly authoritative sources, though.