Joe Root contributed a magnificent 160 to England's first-innings tally of 384 but Travis Head drove Australia to 166 for two in response on the second day of the fifth and final Ashes test on Monday.
England's hopes of winning the urn are long gone after defeats in the first three tests of the series but they continued to show the spirit, lacking in previous beaten touring sides, that was evident in their victory in Melbourne last week.
Until Australia took the second new ball, Root looked in full control in the Sydney sunshine on a pitch that looked as benign as the surface for the chaotic Melbourne test was fiery.
In a string of partnerships which might have transformed England's fortunes when the series was still alive, Root crafted an innings of real quality from 242 deliveries with 15 fours and no sixes.
Rarely approaching anything close to flamboyance, he punched two runs back past the bowler to reach his 41st test century and second in Australia after his hundred in the second match in Brisbane.
"Looking at the surface and how it could potentially play out here, maximizing that first-innings score was always going to be important," Root told reporters.
"To contribute heavily to that is obviously what it's about as a batter, that's your job. There's so much to play for throughout the next three days and we find ourselves very much in the game."
Michael Neser (4-60), the pick of Australia's bowlers, ended Root's knock with an athletic catch off his own bowling and two deliveries later bowled Josh Tongue to terminate England's innings and bring up an early tea.
It was England's highest tally of the series but they needed their bowlers to back it up in a marathon final session if they were to have a chance of going home 3-2 losers.
The bowlers created some early chances with Jake Weatherald dropped twice in the space of three balls -- forgivably perhaps for Root in the slips, not so much in the case of Ben Duckett at square leg.
Weatherald was trapped in front by Ben Stokes for 21 and the England captain also removed Marnus Labuschagne for 48 with stumps in sight.
Head, one of only three centurions in the series along with Root and Alex Carey, proved impossible to budge, though.
He reached the half-century mark for the third time in the series with a typically aggressive shot through the covers and will resume on day three on 91 not out alongside nightwatchman Neser, who had scored one.
"It's always a winner the longer we bat, especially because you're keeping them out on the field," said Neser.
"The wicket seems to be quickening up just a little bit, so we've got to be mindful of that tomorrow."
Root-Brook partnership ended
Earlier, Root had resumed on 72 with England 211-3 but his partnership with Harry Brook was quickly brought to an end at 169 runs.
After controlling his wilder impulses on day one, Brook fell for 84 after a late stab at a Scott Boland delivery which went straight to Australia skipper Steve Smith in the slips.
There was then a cameo appearance from Stokes, who was dismissed without scoring when a DRS review revealed the slightest of touches on his bat and Mitchell Starc claimed a 28th wicket of the series.
Root was soon building a partnership of 94 for the sixth wicket with Jamie Smith, although the wicketkeeper was fortunate to survive on 22 when he was caught off the bowling of Cameron Green only for a no-ball to be called.
Smith holed out for 46 with a poor shot off Labuschagne's occasional bowling just before lunch but Will Jacks (27) and Root put together a stand of 52 for the seventh wicket.
Batting was getting much harder in the face of the new ball and England added only nine runs for their last three wickets.
