England were 28-2 when rain forced play to be abandoned at 1715 GMT, a lead of 35 runs, after Australia were bowled out for 386 before lunch in reply to the hosts' first innings total of 393. Usman Khawaja scored 141 runs - his first test century in England.
When play was first halted opener Zak Crawley was seven not out while Ben Duckett was on 18, but both departed in quick succession after the restart as Australia's fast bowlers found dangerous movement in fast-fading light.
Pat Cummins dismissed Duckett for 19 after Cameron Green pulled off a spectacular diving catch in the slips, his second blinder in as many weeks after his involvement in the dismissal of India's Shubman Gill in the World Test Championship final.
Three balls later, fellow opener Crawley (7) followed him off the field after edging a Scott Boland delivery to wicketkeeper Alex Carey.
"It was sort of the first time the pitch came out a little bit in three days," bowler Ollie Robinson told the BBC.
"If we carried on batting, it might have been a slightly different approach, but still positive... Hopefully we can get the ball moving and bat all day tomorrow. I think anything over 300, we are right in the game."
With rain falling, their replacements Ollie Pope and Joe Root, both yet to get off the mark in the second innings, headed off gratefully when play was once again halted with dark clouds hovering menacingly above the stadium.
England, who have won 11 of their last 13 tests under Stokes and head coach Brendon "Baz" McCullum, are looking to regain the Ashes against world champions Australia in the five-test series.
Australia had resumed on day three at 311 for five on a warm, overcast morning at Edgbaston, but lost both settled batters Khawaja and Alex Carey before three more wickets fell in quick succession as the visitors ended on 386 - seven runs short of their target.
Khawaja played with great composure for most of his knock, showcasing his experience as he picked off the singles and kept the ship steady.
However, as England captain Ben Stokes piled on the pressure with an aggressive field placement comprised of six catchers, Khawaja took the bait, charging down the wicket before Robinson's inch-perfect yorker uprooted his off-stump.
Earlier, James Anderson was denied Carey's wicket in the day's first over due to a Jonny Bairstow drop, but an angling delivery from the veteran bowler snared the Australian for 66 two overs later, ending his 118-run stand with Khawaja.
Australia captain Pat Cummins contributed 38 runs, hitting Moeen Ali for two sixes in an over before the spinner briefly left the field for treatment on an injury to his spinning finger.