The minnows came into the clash having lost four out of four Group D matches, conceding 21 times without scoring.
But that goal drought ended as, with Gibraltar trailing 1-0 to Shaun Maloney's penalty, Lee Casciaro ran onto Aaron Payas' throughball and drilled low through David Marshall's legs to equalise.
Gibraltar's joy was short-lived, however, as Steven Fletcher's hat-trick as well as another Maloney penalty and Steven Naismith's goal saw Scotland run out comfortable winners.
"All the credit should go to Gibraltar today," Strachan said. "They made my life a misery for periods of that game and a long game for me as a coach.
"[Goalkeepers] Allan McGregor and Craig Gordon now love me and think I'm the best manager in the world for not picking them; they're not in the history books.
"It was partly down to us. We spent all week on attacking and width; we then fell asleep taking it for granted they wouldn't attack, and they did.
"We were caught walking out when we should've been walking back the way. So, that made it an interesting couple of minutes.
"But Shaun held his nerve and we've all been there: taking a penalty against one of the minnows, and he dispatched that well.
"It was nice to see Steven Fletcher score a hat-trick and he was one of our better players today.
"It's ironic because I was here [at Hampden] when Colin Stein scored four [in 1969] - that was a long time ago."
