Jimmy Anderson floored Warner with an unintended beamer when Australia were just short of a hundred, with the batter just managing to fend off the shoulder-height ball with his bat handle.
Fast bowler Mark Wood then hit Khawaja on the helmet with a bouncer shortly after, leading to a lengthy concussion check before the rain set in.
It is the highest opening partnership in the Ashes in England since the home side's Alastair Cook and Andrew Strauss put on 196 at Lord's back in 2009.
England's Stuart Broad, who announced his retirement the previous night, and Anderson, marking his 41st birthday, had opened the bowling after ending England's second innings together in the morning, with Broad smashing the final ball of his batting career for six.
Chris Woakes, Moeen Ali, Joe Root and Wood all joined the attack for spells under the floodlights on an overcast day, but were unable to break through the openers' defences.
England had batted with great intent on Saturday, with Zak Crawley (73), Root (91) and Bairstow (78) helping to build the score as Australia's bowlers struggled to restrict the flow of runs.
No team has ever chased down more than 263 at The Oval - a record set by England against Australia back in 1902 - although 10 sides have scored more than 300 on their way to draws at the south London ground.
Play will start again on day five with Australia needed a further 249 wins.
Australia have retained the Ashes after the fourth test was washed out, leaving them 2-1 up in the series. They are hoping to win the series outright however, to become the first Australian side to do so in England since 2001.
