The plot of mixing cards face up and face down, then having them right themselves (with several variations) seems to have first appeared as Theodore DeLand's “Inverto”, 1914, a marketed effect that employed a deck in which every other card was double-backed. See Walter Gibson's article on DeLand in The Conjuror's Magazine, Vol. 1 No. 8, Sept. 1945, p. 12. Also see Ellis Stanyon's description in Magic, Vol. 15 No. 3, Dec. 1919, p. 18, under the title “The Self-Reversing Cards,” and Charles Jordan's ungimmicked version, “The Alternate Reverse”, in Thirty Card Mysteries, 1919, p. 54 of the second edition. Another early example appears in Walter B. Gibson's Twenty New Practical Card Tricks, 1925, p. 16, credited to Herbert Johnson, and includes a reverse Topsy-turvy effect wherein an unmixed deck instantly becomes mixed with face-up cards interspersed between face down. This is also a precursor to the effect of the deck magically shuffling itself.
In Two Dozen Effective Practical Card Tricks, 1927, p. 37, Walter B. Gibson published “The Shuffle Reverse”, which features an apparent riffle shuffle of a face-up half into a face-down half. In Underworld, No. 2, 1995, p. 21, Karl Fulves mentions another version of the plot that appeared in Art Altman's marketed trick “Altman's Upside Down Trick”, 1928 (see the advertisement in The Sphinx, Vol. 27 No. 9, Nov. 1928, p. 440). The original instructions are reprinted in Underworld. This version appears to be the first to add the revelation of a chosen card to the magical righting of the deck.
The name “Triumph” was established when Dai Vernon's version of this trick appeared in Stars of Magic, Series 2, No. 1, 1946, p. 23 of the 1961 compilation book. It quickly became the generic title for the effect often called “Topsy-turvy” in previous times. Vernon's also seems to be the first of the riffle-shuffle variety to employ a false shuffle instead of a secretly reversed portion of the deck.
The first instance of doing Triumph with a freely named card seems to be Roy Walton's “Named-Card Shuffle” in Ibidem, Nos. 34/35, Aug. 1969, p. 827.