It wasn't a classic performance in muddy conditions at Pittodrie, but the home fans saw a team committed to attack.
The winner came from a well-worked first-half move between former Ranger Charlie Adam, who passed a free-kick laterally for Celtic's Charlie Mulgrew to lash home his first international goal.
It came at a ground where the full-back enjoyed several successful seasons for Aberdeen - also a familiar home for the new Scottish manager.
Nine of the starting XI were present or past Old Firm players as Strachan appeared keen to summon up the traditional Scottish aggression sought by many fans.
Mulgrew and Chris Burke both had praise for their new manager after the match.
The Celtic defender said: "It's been good, been different, some new ideas in there."
Mulgrew also hoped the victory would kick-start Scotland, adding: "It was difficult conditions...the most important thing was getting the win so we are happy. It is good to get that winning spirit - hopefully we can go on and get more wins."
Burke was making his first Scotland appearance since 2006 and he said: "It's been excellent, him (Strachan) and his coaching staff...I've learned so much so it's great to be amongst it all. It was excellent, I was happy with my performance - all round the team did well."
Strachan admitted his Scotland managerial debut had sparked feelings he had never experienced in his career and was "blown away" by nerves and excitement in the run-up to the game.
Strachan was back at Pittodrie, where he helped Aberdeen to many memorable victories as a player including a famous comeback against Bayern Munich that helped the Dons on their way to the 1983 European Cup Winners' Cup.
As a manager, Strachan has experienced cup finals on both sides of the border but he admitted nothing could have prepared him for managing his country.
The former Celtic and Southampton boss said: "I'm glad it's all over because in 40 years of football that is probably the most excited and nervous I have been before a game in my life.
"I didn't know how I would feel when I first took charge of the team. Now I know. It was an incredible experience.
"I don't think I expected that. I thought I had done nearly everything in the game, played everywhere, and it kind of blew me away really.
"I've been in the Champions League, I have had to beat AC Milan, Barcelona, Man United, and it wasn't like that."
The former Manchester United and Leeds midfielder admitted he had trouble sleeping the night before the game.
He added: "I think it's when you're on your own at times, when you wake up in the morning and you go 'bang, here we go'. It's okay when you get here.
"I don't know what it was, but it was a bit different."