England lost 5-20 in 44 balls during the final session at Headingley to stagger to 82-5 at the close, still trailing by 261.
Johnson took 3-21 and Hilfenhaus 2-37 after North had scored 110 for his third hundred in only his sixth test.
England, dismissed for a below-par 102 on Friday before bowling Australia out for 445, is now left with an almost impossible task to survive the last three days and hold on to its 1-0 lead in the five-match series as it bids to reclaim the Ashes urn.
England is facing its worst ever home test loss to Australia, currently it is an innings and 180 runs at Trent Bridge in 1989. England still has work to do to avoid its worst overall test loss - an innings and 226-run defeat to West Indies at Lord's in 1973.
"We are disappointed we have made the same mistakes with the bat towards the end of the day as we did in the first innings, losing wickets in clumps," said England bowler Stuart Broad, who earlier claimed a test-best 6-91.
"We started well with Straussy and Cooky and then to lose five wickets for 20 runs is really disappointing, though the Australians bowled fantastically well."
England had batted well to get to 58-0 before captain Andrew Strauss was out lbw to Hilfenhaus for 32 and Ravi Bopara departed the same way next ball without scoring in the 23rd over.
Three overs later, Ian Bell was caught for 3 by Ricky Ponting at second slip off Johnson to make it 67-3. Paul Collingwood was lbw for 4 to an in-swinger from Johnson seven runs later, and Alastair Cook on 30 edged Johnson to the wicketkeeper and it was 78-5.
James Anderson was yet to score and Matt Prior had 4 - and was dropped on the day's last ball by North at third slip off Johnson.
Bopara's top score in the series is 35 from seven innings and his place for the fifth test from Aug. 20 must now be in question, despite appearing to get an inside edge to the ball and also having scored three straight centuries against West Indies this year.
Whereas Australia has made batting conditions appear easy, England struggled again in sunny conditions.
"Going into this test match, it was quite obvious for both teams this was the biggest test of the series to date and we both needed to play well," North said. "We took a lot out of the (drawn third) test at Edgbaston, like England did at Cardiff (when England drew a match Australia dominated).
"We expected to win at Cardiff and it left us deflated, and the same thing happened at Edgbaston when we left with the momentum."
North, who also scored 96 in the last test, brought his 100 up with a six over midwicket from offspinner Graeme Swann. He hit 13 fours in 206 balls and batted for 326 minutes.
North, who came in at 151-4 when the match was still in the balance, extended Australia's lead with a fifth-wicket partnership of 152 with Michael Clarke (93). He then put on 70 from 98 balls with No. 8 Mitchell Johnson, who scored 27.
Clarke missed out on a third century in as many tests when he was trapped lbw by Graham Onions before lunch. He now has 445 runs in the series at an average of 89.
Australia's vice captain batted for 138 deliveries and hit 13 fours. He made 103 not out at Edgbaston in the third test when he and North combined to save the match, 136 in the second match at Lord's and 83 at Cardiff in the first test.
Clarke was untroubled for most of the morning, though a leading edge off Onions when on 92 landed just short of mid-off after he tried to flick the ball to midwicket.
He also edged James Anderson through the vacant third slip area when on 52, but for much of the morning he dominated, driving through cover especially well.
North, on 7 overnight from 42 balls, increased his run rate. His nonchalant slash for four off the back foot through cover from Stuart Broad to go to 83 showed his growing confidence. He was the last man out, caught on the midwicket boundary off Broad.
No. 10 Stuart Clark further humiliated England by scoring 32, hitting Swann for a six before tea. Next over, Clark slapped Broad over midwicket for consecutive sixes before eventually playing on to the same bowler.
Anderson, who appeared to be hampered by injury after hurting his left leg while running a quick single on Friday, did not bowl between lunch and tea.
The first and third tests were drawn, while England won the second match at Lord's.
