Mulgrew, who can play both in midfield and defence, was 'magnificent' against Ireland, Strachan says, while he has come on leaps and bounds during his time with the Bhoys.
Strachan believes the key to the 29-year-old Celtic star's development has been his experience of Champions League football.
Mulgrew is currently recovering from a torn hamstring tendon and may have to wait a while before he can return both for club and country.
But that doesn't mean Strachan has forgotten just how good he can be...
“Charlie was magnificent against Ireland," the Scotland boss said of the Celtic man.
“Charlie, through playing Champions League football, has come on leaps and bounds.
“Some clubs can’t give you that, no matter how good they are or how good the coaching is. It’s invaluable when they come to play in the national side.”
Celtic face Dundee United later today in the Scottish Cup quarter-finals as they aim to secure a historic treble during Ronny Deila's first season at the club.
Next Sunday, they play Dundee United again, only that clash will be contested in the Scottish League Cup final. In the Premiership, meanwhile, the Bhoys are six points clear of second-placed Aberdeen.
Mulgrew will no doubt be raring to get back in among the action when he can. Will his Celtic team-mates deliver come the business end of the season in his absence?