However, their more immediate task was to bat out the day's remaining two sessions and deny holders Australia, bidding for a first away Ashes series win in 18 years, the six wickets they needed to go 1-0 up in the five-Test contest.
Bancroft caught Joe Denly (11) at short leg after Lyon struck again to leave England 80-3 before Root was dismissed.
'I'm absolutely confident we can do that, ' he said.
England started the day needing 398 to win, knowing a draw was realistically the best they could hope for and Rory Burns (11), who became just the fourth England player to bat on all five days of a Test match, fell in the third over of the morning after getting caught in two minds trying to play a short ball from Cummins and gloving it to Lyon at gully. Woakes was the last man out in the 53rd over to Pat Cummins (4-32), sending the ball to Steve Smith in the slips as England collapsed in the afternoon.
England then claimed a first-innings lead of 90 runs and had Australia 2-27, only for Smith to again steady.
When Chris Woakes and Moeen Ali came together at the crease the score England were 97-7.
"Credit to Australia", he added.
Talking about his performances, Smith told BBC, "It feels pretty special, winning the first Test in England is really special".
England's instinctive batting aggression paid huge dividends in its Cricket World Cup title last month, including a semifinal win over Australia in Birmingham, but the switch from white to red ball is proving a tough challenge so far against its well-prepared opponent.
An emotional Steve Smith admitted his brilliant contribution to Australia's first Test humbling of England was a dream come true after the pain of his ball-tampering ban. Been a long 18 month for me, friends and family got me through some hard times and got me back where I am.
Smith changed the tone of the match, leading Australia to 284 all out in their first innings from a perilous 122-8.
"I love playing cricket here in England. It was a relief to get it out of the way", he said.
"We have got the confidence that we can do it [win the series]". "They're creating a good atmosphere for us and we're grateful for it". Australia, meanwhile, had lost 15 successive games on Warwickshire's turf, dating all the way back to 2001.
And it also means England have some work to do and some pressing selection issues to ponder ahead of next Wednesday's second Specsavers Test at Lord's.