Canadian Reynolds, famed for starring in superhero comedy "Deadpool", and American actor McElhenney, creator and star of sitcom, "It's Always Sunny in Philadelphia", first expressed interest in investing in the club in September.
In November, the club's supporters trust (WST), which owns the club, voted in favour of the move.
"We can now pursue our goal to grow the team and return it to the English Football League (EFL) in front of increased attendances, and in an improved stadium, while making a positive difference to the wider community in Wrexham," Reynolds and McElhenney said in a joint statement.
Wrexham were relegated from the EFL in 2008 and are currently seventh in the National League, one division below the EFL's League Two.
The club said the new owners had also invested £2m as part of the takeover deal and that "first-team player identification" would be a priority.
Further funds will be used to enhance the women's programme.
Wrexham, who have reached the FA Cup quarter-finals three times but have never competed in the top-flight of English soccer, were founded in 1864 and have been owned by their fans since 2011.
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