Shaun Maloney's 20-yard curler after a fine corner-kick routine was enough to settle the Celtic Park clash and draw Scotland level on points with Ireland in Group D - ahead of them on head-to-head record.
The goal was a rare moment of quality in a tough Glasgow contest, played out in what Strachan described as a "British cup tie" atmosphere, while the Scotland coach also credited assistant Stuart McCall as the strike's inspiration.
He told Sky Sports: "I am [a proud man] because that was an epic duel.
"From early in the game we knew what was going to happen but we dealt with that most of the time and we tried to get on the ball and play.
"They're a big strong side, they're professional, well organised, it was going to be hard.
"We kept persevering and trying to play and eventually, from a Stuart McCall corner kick we scored a goal.
"You've still got to get in that position and you've still got to be a great player to finish from there, setting the ball outside the goalpost and bringing it back in, so Shaun can be delighted with himself."
Strachan also confirmed that Steven Fletcher will not be risked for next week's friendly clash with England, also at Parkhead, after picking up a knock on his ankle during the game.
He explained: "We're not going to be able to risk him for [the sake of] Sunderland, that's for sure.
"If he's walking out of here with one of the protective boots on we won't risk him for Wednesday."
