Eddie was born in Wahiawa, Hawaii, which is a small town in the middle of Oahu Island, on the 27th of July, 1935. He graduated from Leilehua High School and joined the US Army in 1953. When he was stationed in Okinawa, he met Jane, who was working in the Women's Army Corps, and was also Hawaiian born and they were married soon after. Once stationed in Tacoma, Washington, they had two daughters, Kathleen and Pamela, but Eddie was soon deployed to the Republic of Vietnam. He was the platoon sergeant of 1st Platoon, Company C, 2nd Battalion, 8th Infantry, 4th Infantry Division on the 16th of February, 1967, and it was his actions that day that would earn him the Medal of Honor. The citation reads:

For conspicuous gallantry and intrepidity at the risk of his life above and beyond the call of duty. During a reconnaissance patrol. his platoon was suddenly engaged by intense machinegun fire hemming in the platoon on 3 sides. A defensive perimeter was hastily established, but the enemy added mortar and rocket fire to the deadly fusillade and assaulted the position from several directions. With complete disregard for his safety, P/Sgt. Smith moved through the deadly fire along the defensive line, positioning soldiers, distributing ammunition and encouraging his men to repeal the enemy attack. Struck to the ground by enemy fire which caused a severe shoulder wound, he regained his feet, killed the enemy soldier and continued to move about the perimeter. He was again wounded in the shoulder and stomach but continued moving on his knees to assist in the defense. Noting the enemy massing at a weakened point on the perimeter, he crawled into the open and poured deadly fire into the enemy ranks. As he crawled on, he was struck by a rocket. Moments later, he regained consciousness, and drawing on his fast dwindling strength, continued to crawl from man to man. When he could move no farther, he chose to remain in the open where he could alert the perimeter to the approaching enemy. P/Sgt. Smith perished, never relenting in his determined effort against the enemy. The valorous acts and heroic leadership of this outstanding soldier inspired those remaining members of his platoon to beat back the enemy assaults. P/Sgt. Smith's gallant actions were in keeping with the highest traditions of the U.S. Army and they reflect great credit upon him and the Armed Forces of his country.

Eddie's wife Jane, received his Medal of Honor from Stanley Resor, the Secretary of the Army, on the 3rd of October, 1968. According to the News Tribune from Tacoma, Washington, Eddie had been home on rest and recuperation with his family just days before his death. Elmelindo Rodrigues Smith is buried in the National Memorial Cemetery of the Pacific (also known as the Punchbowl Cemetery) in Honolulu Hawaii; Section W, Grave 131, where his wife would join him in 2002. His name appears on Panel 15E, Line 51 of the Vietnam Veterans Memorial Wall in Washington DC.