Eastwood was born on 12 March 1910, the son of George Eastwood and Eda (née Brooks), the youngest of six children. Eastwood was educated at Oldham Municipal Secondary School. He initially studied physics at Manchester University but switched to education, leaving with a Diploma in Education in 1933. Before he could apply for a job as a teacher he was awarded a grant to study for a PhD at Christ’s College, Cambridge in the department of physical chemistry under C P Snow. He was awarded a PhD in 1935. From 1935-1941, Eastwood worked as a schoolmaster, first at King's School, Taunton, then Runcorn Grammar School. In 1937 he married Edith Butterworth. They moved to Liverpool where his first son John was born (they went on to have another son).

In May 1941, Eastwood volunteered to join the Royal Air Force and was transferred to 60 Group, which was responsible for radar stations and ground control interception of incoming bombers. Eastwood's discoveries about callibration contributed to the development of radar and he achieved the rank of Squadron Leader (mentioned in despatches).

Eastwood joined The English Electric Company in 1946 where he worked on re-engineering wartime radar. The company was bought by the Marconi Wireless Telegraph Company, and Eastwood rose to the position of Director of Research (1954-1962) where his recommendations led to major defence contracts for the company and a range of radar systems. In 1959 Eastwood entered a partnership with the Edward Grey Ornithological Institute, Oxford, using radar to study the flight and migration of birds. This work continued in the 1960's and led to the publication of many papers on radar ornithology. He also worked on developing radar for air traffic control uses and detection of guided missiles.

In 1962 Eastwood was appointed Director of Research at the English Electric Company (1962-1968) then at the General Electric-English Electric Company (1968-1974). From 1974 he was a Consultant for GEC-Marconi Electronics Ltd and for the GEC-Hirst Research Centre.

Eastwood also worked on government and professional bodies. He chaired the Radar Committee of the Electronics Research Council, then chaired the Electronics Research Council (1970-3) and was a member of the Defence Scientific Advisory Council (1969-1972, 1976-9) and of the Science Research Council (1968-74). He was President of the Institution of Electrical Engineers (1972).

Eastwood received many honours for his achievements including the Wakefield Gold Medal (Royal Aeronautical Society, 1961), the Glazebrook Medal (Institute of Physics and Physical Society, 1970) and the Sir James Alfred Ewing Medal, Institute of Civil Engineers, 1976). He was made CBE in 1962 and elected FRS 1968, and he was knighted in 1973. Eastwood's publications included various papers on spectroscopy, radar techniques, radar meteorology, radar ornithology as well as "Radar Ornithology" in 1967. Sir Eric Eastwood died on 6 October 1981.