The same thing was done in the U.S., where malaria used to be endemic until the 20th century. http://dcmosquitosquad.com/history-of-malaria-in-the-usa/
"If malignant malaria were eradicated from human populations, chimpanzees, in addition to gorillas, might serve as a reservoir for P. falciparum."
Basically to get rid of it you have to kill the mosquitos or stop the people getting infected. That is hard with large jungle areas and tribal people who don't go see the doctor. In most places where people go to the doctor it does get wiped out. Eventually as Africa gets richer and people have modern facilities it would probably get sorted anyway but that may be a while. Things can go backwards too. In Venezuela it was mostly gone but has come back with the economic collapse.
The climate is probably not the best for the parasite, but there was a number of changes that more or less unintentionally helped erradicate the disease.
First, there were enormous projects to improve farming. Basically all lakes became regulated and most wetlands were dried out to gain farmland. Before that there was a long period of improvements of living standards and hygiene - as simple as separate and better houses for the people rather than sharing a single house with the kettle. Glass windows got affordable for basically everyone, etc.
There are still mosquitos, including those that may transmit malaria, but the breeding opportunities and contamination paths got reduced below a critical level for the parasite.
I suspect that having cold winters helps, but I guess that the situation in Africa could be improved too with better houses and better control of potential breeding areas. It took a couple of hundred years in Sweden though..