Dawid Malan’s 140 and Bairstow’s 119 were the high points on a fluctuating second day for the tourists, who headed into the game 2-0 down in the five-match series. England lost their last six wickets for 35 in their first-innings 403 all out.
Steve Smith’s 92 not out and Usman Khawaja’s 50 then helped Australia reach 203-3 at stumps.
“(On Friday) morning it was pretty tough out there,” said Bairstow. “But I’m delighted to get my first Ashes hundred. It’s something as a boy you really want to do. I’ve been fortunate enough to play in a few Ashes series now - to get it at the WACA, away from home, means a heck of a lot.”
The 28-year-old wicketkeeper-batsman completed his hundred in 185 balls, triggering a cheeky celebration as he followed several punches of the air with a headbutt for the helmet he had just taken off.
The reference to his unconventional greeting for Australia opener Cameron Bancroft, in a Perth bar seven weeks ago, was lost on no-one in the ground and Bairstow enjoyed the moment.
“There’s always some feeling, just a bit of light fun, and that’s exactly how it should be taken,” Bairstow said of the celebration. “(There’s a) huge amount of passion within the game - that’s been throughout the series.”
Asked whether England had targeted more than 403, he said: “Potentially we could have kicked on and got a few more than that. It’s something to say we got 400. After being 100-4, I think we’d have taken that.
“They played pretty well (on Friday) evening. (Saturday) morning’s a crucial session for us, to try to keep them out there for as long as possible and scoring at a rate we’re comfortable with.
“At end of the day (on Saturday), who knows what could have happened? They obviously took six wickets in the morning session - there’s no reason we can’t. The wicket’s offering a decent amount as well.
“If we get it in the right areas, with the skills the boys have shown throughout the series, there’s no reason why we can’t do the same.”
Bairstow said England will head into day three in confident mood.
“It’s difficult to compare the two attacks because obviously the skills are very different, the paces are different,” he said. “But what I do know is that it’s done enough for us to get 20 wickets in this Test match. That’s the exciting thing about it.
“Over the next couple of days it might offer a bit more turn. Mo (Moeen Ali) I thought bowled delightfully, using the wind, using the drift, and the control he had... It was probably the best I’ve seen him bowl for a while.”
Khawaja too was upbeat about his side’s chances, if Smith and Shaun Marsh can stay at the crease for a while into the weekend.
“The WACA’s a beautiful place to play, there’s a lot of value for shots out here,” he said. “If you can get a partnership going, runs can flow very quickly.
“Steven Smith’s batted beautifully - hopefully he can keep going and obviously Shaun Marsh is in terrific touch too so hopefully we can get a nice little partnership there starting (on Saturday).”
Khawaja felt his side had done well to keep the tourists down to 403, noting: “Malan and Bairstow batted beautifully and they started well (on Friday) morning too... It could have been a lot worse for us but our bowlers bowled really well.”