Rodgers has been appointed the new Celtic manager following the departure of Ronny Deila.
The former Liverpool boss has been handed a 12-month rolling contract by the Scottish Premiership giants.
Smith believes that Rodgers will hit the ground running at Celtic, who have won the league title in Scotland for the past five seasons.
The Scotsman has also hailed the signing of Joey Barton by Glasgow giants Rangers.
The Gers, who won the Scottish Championship title with ease this past season, have secured the services of the former Newcastle United and Queens Park Rangers midfielder on a free transfer.
Barton is expected by many to play a key role in Rangers’ quest to win the Scottish Premiership title during the 2016-17 campaign.
“He (Rangers manager Mark Warburton) certainly has issues to be addressed,” Smith wrote in The Sunday Post. “The defence isn’t good enough and will need to be reinforced, as will other areas of the team. Certainly the arrival of Joey Barton will help.
“Always a good player, he has added maturity and discipline. He stayed suspension-free as Burnley won the English Championship, despite being a near ever-present.
“Rangers will need him, and more, because Brendan Rodgers is, I’m sure, going to hit the ground running at Celtic. I know Brendan. He leaves no stone unturned."
Rodgers is a good manager and an intelligent coach, and he should be able to succeed at Celtic.
The Bhoys’ main target at the moment is to qualify for the group stage of the Champions League next season.
The former Liverpool manager will have to identify potential signings who will help the Bhoys achieve that feat.