England were 200-5 when bad light ended play shortly before the scheduled close, still 297 behind Australia's imposing 497-8 declared built on star batsman Steve Smith's commanding double century.
Fast bowler Hazlewood has so far taken 4-48 in 20 overs, with England now needing a further 98 runs just to avoid the follow-on.
Ben Stokes, whose unbeaten century took England to a stunning one-wicket win in the third Test at Headingley, was seven not out and Jonny Bairstow two not out.
England were making steady progress during a third-wicket partnership of 141 between Rory Burns (81) and captain Joe Root (71).
But Hazlewood removed both well-set batsmen before bowling Jason Roy in a superb burst of 3-15 in 27 balls.
Hazlewood's haul followed on from his nine-wicket match return at Headingley, with the 28-year-old quick arguably in the best form of his injury-plagued career.
"It's up there," he told reporters after stumps. "I've been consistent in the last two-and-a half Tests. The body feels fresh.
"I am bowling a lot of deliveries at off stump and on the pads due to my action. It's only a little change but I have worked hard on it."
Hazlewood was ably assisted by fellow paceman Pat Cummins, unlucky not to take a wicket on Friday.
"Patty rarely bowls a bad spell to be honest," Hazlewood said. "He just gets better every time he bowls."
Defeat in Manchester would end England's chances of regaining the Ashes at 2-1 down with one to play in a five-Test series.
"We probably wouldn't have liked to lose a couple of wickets at the end but I think we are only a couple of partnerships away (from avoiding the follow-on)," said left-handed opener Burns after his third score of fifty or more this Ashes.
Rain meant there was no play before lunch.
England resumed on 23-1, with Burns 15 not out and nightwatchman Craig Overton three not out.
Overton had added just two to his score when he edged Hazlewood to Smith at second slip.
Burns, who scored his maiden Test century in the series opener at Edgbaston, cover-drove Mitchell Starc for four and a steered boundary off the left-arm fast bowler, playing his first match of the series, saw him to a 100-ball fifty.
- Root's reprieve -
===================
Root should have been out on 54.
But an edge off Cummins went between Australia captain and wicketkeeper Tim Paine -- who should have gone for the catch -- and David Warner at first slip.
Hazlewood and Cummins kept the pressure on from both ends, each achieving lift and movement in a way that only Stuart Broad had managed when Australia batted.
The Australia duo's display was a reminder of how much England missed James Anderson after their all-time leading Test wicket-taker, who has been sidelined since the first match with a calf injury, was ruled out of the rest of the series.
Burns, in sight of what would have been a second Test century after his 133 in the series opener at Edgbaston, fell when he was undone by a fine Hazlewood ball that nipped away and took the edge, with Smith holding another sharp slip catch.
"It was disappointing but it wasn't a bad ball either," said Burns, who batted for nearly five hours.
"It was one of those when sometimes you walk off and you can be pretty at peace with yourself."
Hazlewood then had Root lbw with a fine nip-back ball.
New batsman Roy had been dropped down the order after struggling as a Test opener.
The World Cup-winner got off the mark with a cover-driven four off Hazlewood but the paceman had the last word, clean bowling him middle stump for 22 with a superb delivery that moved late off the seam.