The process in which an existing DNA strand is extended continuously in a 5' to 3' direction by activities including the addition of nucleotides to the 3' end of the strand, complementary to an existing template, as part of DNA replication. Leading strand elongation proceeds in the same direction as the replication fork. [ https://www.worldcat.org/search?q=bn%3A071673706X https://www.worldcat.org/search?q=bn%3A0815316194 ]