Cameron replaces Charlie Gardiner, who will relocate to Melbourne for family reasons after five years in the role.
Gardiner will remain employed by the Swans in a new position focused on legal compliance and commercial projects.
Cameron coached the Giants for 193 games, including a grand final in 2019, following a distinguished 256-game playing career with the Western Bulldogs and Richmond.
After leaving the Giants in 2022, the 51-year-old has spent the past year as technical director of the Swans' academy.
"Leon's experience across all aspects of the game is unmatched and we are fortunate to have someone of Leon's character and skill set transition into the role," Swans CEO Tom Harley said.
"Having been at the club for the past 12 months, Leon has forged strong relationships across the entire club, and I am certain that he will continue to positively impact our organisational performance.
"I am excited about what the men's team can achieve in season 2024."
McVeigh steps up as an assistant to John Longmire after coaching the Swans' academy in 2023.
He also coached the Allies to their first under-18 national championship.
The 42-year-old was an assistant coach to Cameron at GWS and took over as interim coach when the latter stepped down midway through 2022.
McVeigh, who played 232 games for Essendon, had been keen to land the Giants' senior coaching role last year but missed out when Adam Kingsley was appointed.
The latest Sydney football department changes come after assistant coach Don Pyke left to take over as chief executive of West Coast.
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