A , usually employed as a left-back, Edinburgh started his career as a trainee with Southend United, turning professional in August 1988. He helped the Shrimpers win promotion from the in 1990. In all competitions he played 47 games for the seaside club, scoring one goal.
He moved to Tottenham Hotspur for £150,000 in January 1990, initially on loan. He joined Spurs permanently in July 1990, making his debut for them on 10 November 1990, as a in a 4–2 win at home to Wimbledon. He played for Spurs for most of the 1990s. He won an FA Cup winner's medal in despite just previously starting only 12 league games. He would also win a League Cup winner's medal with Spurs, in . This second medal was awarded to Edinburgh despite his being sent off in the final for raising his arms to Robbie Savage, as the post-match analysis showed that Savage had distinctly over-reacted to a minimal contact and that the red card was unjustly awarded. In total, he made 213 league appearances for Tottenham, scoring once.
Edinburgh left London for Portsmouth in March 2000 for a fee of £175,000, playing 35 league games over the next two years, scoring once against Sheffield United. However, he did not play a single competitive game during Portsmouth's Division One title winning campaign (), and joined non-league Billericay Town as player-manager in July 2003.