The Australians were 30-1 at stumps on an intriguing day two that helped to make up for the washout on the opening day of the test.
With England stuttering at 138-6 after being put in to bat, Bairstow put on 72 for the seventh wicket with Chris Woakes (32) before overseeing more valuable runs from the tail which nudged the team to a respectable total.
Bairstow was eventually dismissed for 52, giving Nathan Lyon a third wicket of the innings to move the spinner to 355 in tests — tying paceman Dennis Lillee in third place on Australia's all-time list. Quicks Josh Hazlewood (3-58) and Pat Cummins (3-61) also took three wickets apiece.
Opener Rory Burns was England's top scorer with 53, following up his century in the first test won by Australia by 251 runs at Edgbaston.
It left Australia with an hour to get through and Warner (3) again fell quickly, removed by England fast bowler Stuart Broad for the third time in this series. The left-handed opener, who was clean-bowled, managed only 2 and 8 in the first test.
Fellow opener Cameron Bancroft survived and was unbeaten on 5 off 35 balls, and was in the middle with Usman Khawaja (18).
Australia trailed by 228 runs.
Recalled in place of fellow paceman James Pattinson, Hazlewood was the pick of the bowlers with his consistent line and length, and took the first three wickets of England's innings.
Having removed Jason Roy (0) off the third ball of his first over and then trapping Joe Root (14) lbw also before lunch, Hazlewood took the first wicket of the second session by finding the edge of Joe Denly (30) to give wicketkeeper Tim Paine one of his four catches of the day.
Bancroft then sent Burns back to the pavilion with a stunning catch at short leg, diving low to his left and managing to hold on to the ball with his fingertips off a delivery by Pat Cummins. Burns was extremely watchful, making his 53 off 127 balls.
Jos Buttler and Ben Stokes, England's dangerous middle-order batsmen, went cheaply for 12 and 13, respectively, to leave the hosts really struggling at 138-6 before Bairstow and Woakes launched the rebuilding job.
Roy, a white-ball specialist who was one of the star performers in England's World Cup-winning campaign this summer, is playing only his third test, after making 5 and 72 in a warmup against Ireland, and 10 and 28 in the first test against Australia at Edgbaston.
His place in the team — at least as an opener — might come under scrutiny after lasting only three balls on Thursday, wafting at Hazlewood's first ball and then pushing hard at a delivery outside off stump two balls later to edge to Paine.
Warner is also struggling for runs and he was again snared by Broad, who steamed in from around the wicket and beat the opener for sheer pace.
Broad had 1-13 off five overs.
England gave a test debut to Barbados-born fast bowler Jofra Archer, who replaced the injured James Anderson, and he had six overs before stumps.
Lord's was awash with red in support of the Ruth Strauss Foundation, a lung cancer charity set up to honor the late wife of Andrew Strauss, the former England captain and director of men's cricket. Fans were asked to wear red and both teams wore limited-edition caps and shirts to be auctioned off, one of several fund-raising initiatives in place.
Strauss' two sons rang the five-minute bell before play started.