Welsh began his career with Dundee United, making his debut in the 1986–87 season. During that season, Welsh was part of the Scotland under-19 team who reached the semi-finals of the UEFA European Under-19 Football Championship, losing 1–0 to Italy. The following year, he reached the quarter-finals of the World Youth Championship, losing on penalties to West Germany. Welsh became a regular in the United side in 1993–94 and helped the club win the Scottish Cup for the first time, particularly by scoring in the semi-final against Aberdeen. He then experienced relegation in 1995 and subsequent promotion with the club. Welsh scored a crucial goal in a promotion play-off match against Partick Thistle in 1996.