I doubt that "sufficere" ever meant "put under"; I'll assume that this meaning was just suggested as a crude literal translation rather than attested as real Latin usage.
The same Indo-European morpheme that entered Latin as "sub-" is actually also reflected in English "up" and "above" (and in Sanskrit "upari" meaning "above", "on", "upward", "over" - and indeed, in Latin "super" meaning "above"), so we shouldn't rush to conclude that the the most well known meaning "under" is necessarily the only or original one. There's a fascinating range of other meanings that share a local, temporal, or metaphorical notion of close proximity.
The temporal meaning of "sub" (if used with the accusative) means "just before" or sometimes "just after". This (latter) prepositional meaning of prepositional "sub" could perhaps have been the basis of the corresponding prefix in the relevant forms of "sufficere".
A nice example usage is "resulting from a by-election": if a "consul ordinarius" dies or is deposed, he is replaced by a "consul suffectus". A consul suffect succeeding a regular consul is not quite like the regular consul, in terms of prestige, but they can serve in a similar capacity. That's where you have the intermediate meaning of "cause to take the place of".
However, this is just a later example of an earlier, more general and rather productive usage of the prefix "sub-" in the meaning of "somewhat". Example given here: "subhorridus" meaning "somewhat rough": or, if you will, "adjacent to rough" in the abstract space of degrees of roughness.
The second semantic shift from "somewhat like" to "enough" seems to have been accomplished by the notion that "if you substitute X for the needed Y, X will meet the need Y". If it wouldn't, it wouldn't have been a successful substitution, would it?