Google shows this article about eradication of malaria in Corsica after WWII: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2927611/ . The mosquito population was suppressed by draining marshes and by spraying breeding grounds with insecticides such as DDT. If you disrupt the mosquito population for a few years, the malaria parasite will die out since it needs to be transmitted to humans or other animals as part of its life cycle.

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/