Leroy Lita netted twice with Dave Kitson, Gary O'Neil and Mark Yeates also scoring in a resounding triumph.
Strachan said: "I'm going to enjoy this victory. I saw it coming because we had played well against Peterborough and Crystal Palace.
"We created chances in those games and in this one we scored with our chances. Goals lift you and deflate the other side.
"Our target has been to get into the top six and if you can do that, you can then see if you can catch the big boys."
Yeates rattled the bar from 30 yards before on-loan striker Kitson steered O'Neil's free-kick into the corner of the net.
Boro went further ahead four minutes into the second half. David Wheater's header from O'Neil's left-wing corner was handled by Ben Watson and Lita stepped up to smash home the resulting penalty.
That prompted Rangers manager Jim Magilton to make a double substitution, sending on Rowan Vine and Patrick Agyemang.
Within a minute Vine found space on the left and crossed for Agyemang to score from close range.
But any prospect of a comeback was crushed when Mikele Leigertwood gave the ball away and Lita cut in from the left before shooting past goalkeeper Radek Cerny.
O'Neil made it four by scoring from the rebound after Lita had fired against the post and Yeates completed the rout with a 25-yard drive which flew into the bottom corner.
Strachan added: "I wasn't surprised at Lita's performance given the amount of energy he puts in and Isaiah Osbourne produced a real man's performance in midfield and showed everyone what a good player he is.
"QPR can count themselves a bit unfortunate because the ball went for us today and we scored at crucial times.
"But we've worked hard for that. I believe that if you work hard you will eventually get something and it's finally paid off."
The result was a major disappointment for Magilton, whose team have won once in six matches.
Magilton said: "We were guilty of giving the ball away cheaply and we came up against a good side who really clicked in this game.
"If you give the ball away against good sides, you'll be punished. We've handed out a few beatings this season and now we have to learn from this beating.
"It's all about learning lessons. It's wonderful to play expansive football and get the plaudits, but it sometimes takes a result like this to learn about your character.
"Let's see what we've got against Watford on Monday night. I will expect a response and so will the supporters - and so too will the players.
"This was, I hope, part of a learning curve for this group of players. We certainly need to do better."