I have published a number of blog posts about early phylogenetics involving horizontal gene transfer (HGT). The historical issue is that all of the early publications about HGT of individual genes were about mechanisms and evidence, rather than about the phylogeny, and so explicit network illustrations were rare. It is therefore difficult to pinpoint the first illustrated network.
For example, if we consider HGT to be a subset of genome transfer (or genome fusion) then the first explicit phylogenetic network illustrating this was by Constantin Mereschkowsky (1910) Theorie der zwei Plasmaarten als Grundlage der Symbiogenese, einer neuen Lehre von der Entstehung der Organismen. Biologisches Centralblatt 30: 278–303, 321–347, 353–367 (see The first gene transfer network). However, HGT is conventionally treated as involving a small collection of genes, not whole genomes.
Alternatively, if we consider unrooted phenograms to represent HGT networks, then the first explicit illustration of relationships based on individual genes was by Dorothy Jones & Peter H. Sneath (1970) Genetic transfer and bacterial taxonomy. Bacteriology Reviews 34: 40-81 (HGT networks). However, phenetics is not really phylogenetics.
It seems that if we insist upon an illustration showing a rooted phylogenetic network, then we must turn to the paper by Raoul E. Benveniste & George J. Todaro (1974) Evolution of C-type viral genes: inheritance of exogenously acquired viral genes. Nature 252: 456-459. The summary of this paper is:
Genes related to the nucleic acid of an endogenous domestic cat C-type virus (RD114) are found in the cellular DNA of anthropoid primates while many members of the cat family Felidae lack these sequences. Endogenous viruses from primates are thus concluded to have infected and become part of the germ line of an evolutionarily distance group, the ancestors of the domestic cat.The authors discuss HGT explicitly in the context of a phylogeny:
When the virogenes of two species are more closely related to each other than are the cellular genes, one must suspect horizontal transmission and subsequent perpetuation of the viral genes through the germ line. Figure 3 shows models which could account for the data.
There are three distinct phylogenetic models in this figure, and the third one has three alternative possibilities. The authors conclude that "model cII is most likely." This then appears to be the first HGT network that fits the conventional specifications.