Celtic signed Lennox from Scottish Junior team Ardeer Recreation on provisional forms in 1961 at the age of 18, and he made his first team debut the following March. He went on to score 273 goals in all competitions, second only to Celtic’s all time top scorer Jimmy McGrory’s total of 468 goals. Of his 273 goals, 167 were scored in the Scottish league, making him the fifth top league scorer in Celtic. He won eleven League medals, eight Scottish Cup medals, and five League Cup medals (scoring 63 goals in the competition) and was a member of the 1967 European Cup winning Celtic team, known as the Lisbon Lions, who defeated Inter Milan 2–1 in the Estádio Nacional stadium in Lisbon, Portugal.
He scored in 13 consecutive league games (21 goals) between 2nd March 1968 and 7th September 1968, a record only beaten by Evelyn Morrison of Falkirk in the 1928/1929 season, who scored in 14 consecutive league games and equalled by Finn Dossing of Dundee United in the 1964/1965 season.
He played in a second European Cup final with Celtic in 1970, losing 2–1 after extra time to Feyenoord Rotterdam of the Netherlands at the San Siro stadium, Milan. He was an extremely fast winger and was known by fans as ‘Buzz Bomb’ or ‘Lemon’ as they thought he made defenders look like ‘suckers’. He left Celtic in March 1978, and moved to the United States to play for Houston Hurricane in their debut season in the NASL. After a disappointing 3 goals in 30 games for a struggling team, he got a surprise offer to rejoin Celtic in September 1978. It was a good move, as Celtic took the League Championship that year and the Scottish Cup in 1980. He was the last Lisbon Lion to retire as a player when he joined Celtic’s coaching staff in November 1980.