Weeks was born in Mount Clemens, Michigan, where he attended the public schools and learned the printing trade. He studied law and was admitted to the bar in January 1861. During the American Civil War, he served in Company B, Fifth Regiment, Michigan Volunteer Infantry, and was first sergeant of the company. He became first lieutenant and adjutant of the Twenty-second Michigan Infantry in 1862 and captain in 1863. He was appointed assistant inspector general of the Third Brigade, Second Division, Reserve Corps, Army of the Cumberland, in 1863 and was mustered out in December 1863.

After the war, he was proprietor and editor of a Republican newspaper in Mount Clemens and commenced the practice of law in Mount Clemens in 1866. He served as prosecuting attorney 1867-1870 and then as judge of probate of Macomb County, 1870-1876. He was an unsuccessful candidate for election in 1884 to the Forty-ninth Congress, but was elected as a Republican from the 7th District of Michigan to the Fifty-sixth and Fifty-seventh Congresses, serving from March 4, 1899 until March 3, 1903. He was chair of the Committee on Elections No. 3 in the Fifty-seventh Congress. He was an unsuccessful candidate for renomination in 1902, losing the Republican primary election to Henry McMorran, who went on to be elected to fill Weeks's seat in the House. Weeks resumed the practice of law and died in Mount Clemens where he is interred in the Clinton Grove Cemetery.

Edgar Weeks' cousin, John Wingate Weeks, was a U.S. Representative and U.S. Senator from Massachusetts, and U.S. Secretary of War under Warren G. Harding.