Stevens has become Villa's first capture of the January transfer window, although he signed a pre-contract last summer to make the move to the midlands club on a three-and-a-half-year deal.
The Republic of Ireland Under-21 international was allowed to remain with Shamrock for the first half of the campaign and competed in the Europa League.
McLeish said: "It will be nice to have another body in the squad and Enda is a player who has got some potential.
"He's a young player. There was a wee bit of interest in him and we decided to sign him.
"It is up to Enda to hit the ground running and do well and leave me thinking 'let's get this guy involved.'
"We are hoping will get to grips with everything, adapt very quickly to put pressure on guys like Stephen (Warnock)."
Dublin-born Stevens had spells with UCD and St Patrick's Athletic before joining Shamrock in December 2009 and helping them to win the league title at the end of his first season.
He made his Republic Under-21 debut against Cyprus 10 months ago.