Born in Broughty Ferry on 15 December 1947, Davie was on the Chelsea ground staff as a youngster, but the Stamford Bridge side allowed him to return to Scotland to sign for Fife outfit Newburgh before he was snapped up by the Dons in 1965. His displays in the reserves as a youngster won him favour in the eyes of manager Eddie Turnbull, and in February 1967, he was given his baptism at first team level in a league fixture against Ayr at Somerset Park.

Over the next year or so Davie was given half a dozen short spells in the top team, principally as an out and out attacker. Rather unfairly he was given a rough time by the "boo-boys" among the Pittodrie support for his raw style and lack of composure in front of goal. But Eddie Turnbull refused to be influenced by his detractors and insisted Davie was doing all that was required of him. By the end of the 1968-69 season, and playing in a deeper role, Davie had established himself as a first team regular and his all-action displays began to win over his critics. He became known affectionately to fans as "The Brush". Davie more than played his part in the Dons 1970 Scottish Cup triumph, and was a major factor in the Dons' emergence in 1970-71 as the main threat to Celtic's domination of the domestic scene. He was most effective in midfield where he gave no quarter, besides making the most of his strength and presence in the air. Around goal, he never really threw off his habit of scorning the occasional sitter, but Davie could always be relied on to conjure up a score from the most unlikely of positions. You never knew just what to expect when Davie forced his personality on a game.

Bobby Brown handed Robb his first cap in a make or break European Championship clash against group leaders Portugal in Lisbon on 12 April 1971, leading to the fans tongue in cheek terracing chant "We don't need Eusebio 'cos we've got Davie Robb." He played up front in tandem with Alan Glizean and used his aerial ability to great effect, but as the Eusebio-inspired home side threatened to over-run the Scots in the second half, Davie was pulled back into midfield. In the final analysis Robb was one of the few successes in what was regarded as a poor overall performance by the Scots, who went down 2-0 to relinquish their last hopes of qualification in the tournament. Scotland boss Brown was now under severe pressure as manager, but he displayed his faith in Robb by picking him for Scotland's next fixture a Home International against Wales on a waterlogged pitch at Cardiff on May 15, 1971. The conditions were farcical and contributed hugely to the 0-0 scoreline, but Davie was characteristically the busiest man on the park. In the process he unfortunately picked up a thigh knock that ruled him out for the following midweek fixture at Hampden Park against the Irish. However, he recovered in time to be selected for the Wembley fixture against the "Auld Enemy" on May 22.

Davie started the game in wide midfield and gave his usual wholehearted performance. But he was chiefly remembered for being dispossessed by Francis Lee (unfairly, since he received a "hospital pass") in the build up to England's second goal, scored by Martin Chivers. Late in the game Bobby Brown pushed big Davie up front after taking off the ineffective Hugh Curran, but with the English on top it was a forlorn hope. One June 9 Davie lined up with team mates Bobby Clark and Jim Forrest in the Scotland side that lost a now meaningless European Championship qualifier 1-0 in Denmark. Five days later he gained another cap in identical company in a Moscow friendly won by the USSR 1-0. That game proved to be Bobby Brown's swansong as manager, and unfortunately for Davie, new boss Tommy Docherty had little faith in home based players.

Robb continued to give his all for the Dons, but a serious knee injury, picked up in December 1972 began a nightmare run of injuries. He returned from a cartilage operation in March 1973 but his comeback lasted barely four games before he was out again with a similar injury. The next two years of Davie's career were blighted by injury and the 1975-76 season marked his first near injury campaign for four years. "The Brush" played his part in aiding the Dons to avoid the drop in April 1976 and seven months later he was the toast of Pittodrie when he came off the bench to score an extra time winner in the League Cup Final against Celtic. Davie's final season at Pittodrie was played under old rival Billy McNeill in 1977-78 before he joined Tampa Bay Rowdies for £8,000 in February 1978. At the end of the US season he joined Norwich where he played for six months.