Possibly The Web of Easter Island (1948) by Donald Wandrei.
It was certainly already around in 1977 (and already old, though one's mileage may vary) and features an Easter Island statue on the cover.
When Carter Graham uncovers a strange little image, unlike anything else on Earth, which responds to cosmic forces and laws beyond those known to science, he sets out to unravel that greatest of all mysteries - man and his relation to the universe.
This strange search leads him through terrifying experiences to the final and horrifying realization of the awful consequences of his probings.
More precisely, from the reviews below:
I liked the book, a beautiful mysterious start with the finding of the little green statue in the old graveyard of Isling. Bit by bit the reader discovers alongside with Museum director and archeologist Dr. Graham what this tiny but powerful statue's origin is, and what threat it means to the world. The final chapter was a bit of a disappointment.
Found with the Google query
scifi book easter island statue alien site:goodreads.com/book, then switching to the Images results to search for an Easter Island statue cover.