The originator of the fixed point theorem for DCPOs Pataraia proved in


  "A constructive proof of Tarski’s fixed-point theorem for dcpo's", presented in the 65th Peripatetic Seminar on Sheaves and Logic, in Aarhus, Denmark, November 1997


that in a directed-complete partial order (dcpo) with a minimum element each monotone map has a fixpoint. (For the purpose of this particular theorem, a directed set is a poset in which each finite subset is bounded from above, and a dcpo is a poset in which each directed subset has a supremum.) Dito Pataraia is mostly acknowledged for the new proof method rather than the theorem itself.

Who was the first to actually state and prove this theorem is any way? One can argue that the proof of Markowski's Theorem 9(i) in "Chain-complete posets and directed sets with applications" can be adapted to our situation. Still, the clear-cut statement of the aforementioned theorem is not present in Markowski's paper.
The non-constructive version of Pataraia's theorem is called the Bourbaki-Witt fixed point theorem. I learned it from Davey and Priestley's Introduction to Lattices and Order, and Wikipedia gives the following references:


Nicolas Bourbaki (1949). "Sur le théorème de Zorn". Archiv der Mathematik. 2: 434–437. doi:10.1007/bf02036949. 
Ernst Witt (1951). "Beweisstudien zum Satz von M. Zorn". Mathematische Nachrichten. 4: 434–438. doi:10.1002/mana.3210040138. 


However, as usual, some classically equivalent formulations of this theorem become distinguishable constructively. Andrej Bauer and Peter Lumsdaine have a very nice paper about this, On the Bourbaki-Witt Principle in Toposes, where they examine these issues in detail.