Solkar's father was the head groundsman at Hindu Gymkhana, Mumbai. Solkar used to change the scoreboards for the matches played at that ground.^[1] Anant Solkar, Eknath's younger brother, also played cricket at first class level, representing Maharashtra in Ranji Trophy matches.

During his days as a school cricketer, he toured Sri Lanka in 1964 and captained the Indian schools team against London Schools in 1965–66.^[1] The team included future India players Sunil Gavaskar and Mohinder Amarnath.^[1] He played for Sussex Second XI in 1969 and 1970 and became eligible to play for the first XI, but represented them in only one match.^[1]

Solkar made his Test debut against New Zealand at Hyderabad in 1969–70 and volunteered to field at short-leg. He became the first Indian Test Cricketer to be born post independence. He had a successful series against Australia the same season and against the West Indies in 1971. He was selected to open the bowling along with Abid Ali against England in England in 1971. In the first Test match of that series, he scored 67 and formed a 92-run partnership with Gundappa Viswanath which helped India take first innings lead. In the third Test at the Oval, he returned figures of 3/28 in the first innings, scored 44 runs, and took two catches, thereby played an important part in India's win. In the 1972–73 home series against England, he scored 75 in the first Test at Delhi. He took 12 catches in the five-Test series. He did not play well against England in the away series of 1974, but dismissed Geoffrey Boycott in three successive innings (India vs Yorkshire and India vs MCC – first class fixtures). He scored his only Test century against the West Indies in Mumbai in 1975. Apart from his 53 catches in 27 Tests, he made 1,068 runs at an average of 25.42 and claimed 18 wickets at an average of 59.44.^[1] In the 16 years of his first-class cricket career, he scored 6,851 runs at an average of 29.27, including eight centuries, took 276 wickets at an average of 30.01 and took 190 catches.^[1] In Test Cricket, his job as bowler was to bowl 4–5 overs to take the shine off the new ball as much as possible before the Indian spinners took over. For Mumbai's Ranji team he formed an opening bowlers' tandem with Abdul Ismail. In the 1973 Ranji final, he bowled off-breaks on a notorious turner and took 5 wickets to help Mumbai to a famous victory in a match dominated by the spin bowling of Venkat, V. V. Kumar, and Shivalkar.