Lukic signed for Leeds United as a schoolboy. He learnt from
David Harvey and replaced him, initially in a
UEFA Cup Round One, Second Leg, tie against
Valletta 3 October 1979, and then in the League at
Brighton & Hove Albion on 13 October 1979 and went on to play 165 games for the
Elland Road side. After making a transfer request, Lukic moved to
Arsenal in July 1983 for
£75,000, as a long-term replacement for
Pat Jennings. In March 1984 Jennings was injured playing against
Coventry City at
Highfield Road and Lukic came into the side for four games, making his debut on 7 April 1984 against
Stoke City at
Highbury in a 3–1 win. The following season,
1984–85, on 31 October 1984, Jennings played poorly against
Oxford United away in the
League Cup, where Arsenal were beaten 3–2, and Jennings was dropped for one game and Lukic came in to replace him, and though Jennings returned for the next three games, Lukic came back into the side and played every game for the rest of the season, nailing down the number one keepers spot. In
1985–86 Lukic missed just three games in all competitions and in
1986–87 he clinched his first winners medal and won the
League Cup Final against
Liverpool. In
1987–88 Lukic was ever present playing in all 52 matches Arsenal played that season including the return trip to
Wembley against
Luton Town to defend the
League Cup, but Arsenal lost. However, there were better days to come for Lukic and Arsenal in
1988–89, where Arsenal won the title in the last minute of the game against Liverpool. Lukic played his full part in it as he played every minute of every game that season.
1989–90 was Lukic's last season in his first spell at the club. For the third season in a row he played in every single Arsenal match. Having played 277 times for the Gunners, Lukic rejoined Leeds. He became their most expensive signing in May 1990, to replace the ageing
Mervyn Day, playing a further 265 times. Lukic won a second League title in
1991–92 and a runners up medal in the
League Cup in
1995–96 where Leeds lost to
Aston Villa at Wembley. Manager
Howard Wilkinson signed
Nigel Martyn from Crystal Palace in the summer of 1996 and Lukic rejoined Arsenal on a free-transfer in July 1996. Lukic was granted a
Testimonial by Leeds United and his two teams, Leeds United and Arsenal, met at Elland Road on 31 August 1996, in which he played for Arsenal in the first half and Leeds in the second half. As deputy to David Seaman, he played 15 league games in
1996–97 as cover, but after the signing of
Alex Manninger in 1997 he stepped down to No. 3. However, with Manninger injured, Lukic was on the bench for Arsenal's
2000 UEFA Cup final loss to
Galatasaray, earning him a runners-up medal.{{cite news In 2000, after a series of injuries to the club's other 'keepers, he made a brief return to the first team, playing four times. The last of these four games, against Derby County,{{cite news ==International career==