After losing the toss, England restricted the home side to 245, scored 620-5 declared in their innings and then took second-innings wickets with impressive regularity to complete a first victory by an innings against Australia in 24 years.
"Our batters are getting big scores which is crucial and that's putting pressure on the opposition," Strauss added.
"Bowling them out for 245 on a flat wicket was an outstanding effort from Jimmy Anderson [who took 4-51] and the rest of the lads and then we really made hay with the bat.
"We didn't let them back into the game and it was a pretty comprehensive victory.
Strauss, though, was wary about the team becoming complacent.
"We're very happy with what we've done, we're still pretty conscious that there is a lot of hard work to do but it's a nice position to be in at the moment," he continued.
England take the final Australia wicket in Adelaide
"The game of cricket is an odd one and just when you think everything is going your way it can come back and bite you so we've got to keep our work ethic high.
"There is a lot of confidence in the group and that's a fantastic position to be in but if we start getting a little bit complacent, thinking it is just going to happen, that's going to allow Australia back in the series."
With England needing to take six wickets on the final day to secure their victory there were concerns that the dramatic storms predicted might save the hosts, but instead the Adelaide Oval was bathed in morning sunshine.
"I opened my curtains with a huge amount of trepidation, we weren't sure how much play we were going to get, we hoped to get at least two sessions," reflected Strauss, whose decision not to declare overnight after the third day with England already 306 runs ahead was criticised by some observers.
Victory an incredible feeling - Pietersen
"[The weather] was probably going to be the only thing that would stop us from wrapping it up.
"There was enough going on in that wicket we just had to be patient. It was great that Swanny [Graeme Swann] got five wickets. He bowled exceptionally well and kept pressure on in the first innings and then got his rewards in the second."
After battling back from a tough opening day to draw the first Test in Brisbane, England made a dream start to the second match when Jonathan Trott ran out Simon Katich with a direct hit, the first of two wickets to fall in the opening over after Australia had chosen to bat in seemingly near-perfect conditions.
"You get opportunities and they're gone in the blink of an eye and if you don't take them you spend the rest of the day regretting them," Strauss stated.
"That gave us some early momentum then Jimmy followed it up with two outstanding deliveries for [Ricky] Ponting and [Michael] Clarke and then you're off and running."
Strauss also picked out Kevin Pietersen for special praise, the batsman hitting a Test career best 227 - his first Test century since March 2009 - and also taking the key wicket of Clarke with the final ball of the fourth day.
To take a 1-0 lead in the second match of the Ashes is an amazing feeling
"It was always coming," suggested Strauss of Pietersen's innings.
"The way he was playing in the warm-up games and even in Brisbane [where he hit 43] you could see the confidence was there and he loves a stage.
"He's one of these guys who can really take the game away from the opposition and almost humiliate bowlers when he's at his best.
"It was a massive innings for him, and he's full of confidence as are quite a few of our batters."
Strauss, though, said he was confident that either Chris Tremlett, Tim Bresnan or Ajmal Shahzad, who are all expected to play in the tour match against Victoria on Friday, can act as an able deputies.
"We've got some good depth, those are three good back-up bowlers so those guys will be fighting for his [Broad's] place and they come into a confident side which is a good thing," Strauss commented.
Meanwhile, Pietersen admitted it was a very different experience at Adelaide to that of 2006, when former captain Andrew Flintoff and his team lost by six wickets despite making 551 in their first innings.
"Four years ago it was a very sad dressing room, not as united as we are at the moment, not as happy as we are at the moment," Pietersen said.
"To take a 1-0 lead in the second match of the Ashes is an amazing feeling. We didn't think they'd fall over as quickly as they did but if we keep staying in our bubble we'll be successful on this trip."
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