Sent in to bat on a grassy Melbourne Cricket Ground pitch on Sunday, England were skittled for 185, with captain Root (50) alone able to produce a half-century against the hosts’ attack.
England have failed to reach 300 in the series, with their highest first-innings total 236 in the second Test at Adelaide where they lost by 275 runs.
“When you look at the toss that obviously didn’t go our way,” Bairstow told reporters.
“We’d have also looked to bowl in that first session.
“We’re still searching for that big score but both teams were looking to bowl first on that pitch this morning with the help of the conditions and a tinge of green.”
Trailing 2-0 after also losing in Brisbane, England need to win in Melbourne to keep the five-Test series alive.
Australia, meanwhile, need only a draw to retain the urn.
They were 61 for one at the close of day one, with Marcus Harris 20 not out and nightwatchman Nathan Lyon yet to score.
Only one team has overhauled a 2-0 deficit to win the Ashes — Don Bradman’s Australia on home soil back in 1936/37.
Root, all-rounder Ben Stokes and wicketkeeper Jos Buttler all fell to poor shot choices, ending England’s hopes of salvaging a respectable total after Australia captain Pat Cummins tore through the top order with three wickets.
“We’ve got to get a bit stronger and tougher with our dismissals,” said Bairstow, who battled to 35 before caught trying to duck a short ball by Mitchell Starc.
“We know that, we’ve spoken about that.
“Naturally, the execution wasn’t there today but the amount of times those shots are executed and go to the boundary is high.
“People have analysed techniques and game plans as to what’s the best way of scoring runs but at the end of the day it does come down to spending time out in the middle.”