I'm not sure, but I wonder if it's a holdover from days when callers needed operator assistance to place a call.
Because callers needed help calling a place, the language place a call would naturally evolve. If I picked up the phone, and my friendly neighborhood operator was on the other end, "I'd like to you to help me place a call" seems like a perfectly natural thing to say, rather than "I"d like you to help me call a place." (That's just a theory, but the blue line on the graph would seem to support it, as it coincides with the spread of the telephone into rural areas.)
As to why make a call is so much more widely used in the Ngram, not all references to make a call revolve around a telephone. A referee can make a call on the field. Because make a call is an idiom for make a decision, a board of directors can make a call about a new stock option.