Australia reached 154-6 in their pursuit of the daunting target of 267 on an abrasive surface posing all sorts of problems to the batsmen.
The match surely headed for a draw when Australia were 132-3 in the final session but they soon lost three wickets for 17 runs, giving England an outside chance to possibly complete a remarkable victory.
However, England will rue the five sessions lost to rain throughout the course of the match as they lost the race against time failing to get Australia all out in the 48 overs they could deliver after Joe Root decided to declare with his side at 258-5.
Archer's pace bowling heroics saw him register three wickets in the second innings and five scalps overall in his Test debut while Leach returned figures of 3-37.
Australia's star batsman Steve Smith was ruled out of the remainder of the Test with the delayed onset of concussion after he was hit by a bouncer from Archer the preceding day.
Smith's injury might even force him to sit out the Headingley Test, which begins on Thursday.
The 30-year-old's absence could prove costly to Australia given the fact he was the chief protagonist in his side's commanding 251-run win in the series opener at Edgbaston as he contributed with twin centuries.
Smith also scored a vital 92 in the first innings for Australia, rescuing them once again from a precarious position to post a total of respectability.
Former Australian skipper Smith was replaced with Marnus Labuschagne making it the first concussion substitute in cricket history.
Labuschagne, in his debut Ashes Test, was also subject to a barrage of short-pitched stuff from Archer with one of the bouncers striking the grille of his helmet.
He warded off the blows and top-scored for his side with 59.
Earlier, Australia got off to the worst possible start, losing two quick wickets to the fierce pace and guile of Archer.
Labuschagne restored stability to the Australia innings and oozed with confidence with his delectable strokes despite being roughed up by an electrifying Archer spell.
England clawed their way back into the game when Leach had Cameron Bancroft lbw in the first over after tea.
Just when the momentum seemed to shift towards the hosts, Jason Roy was guilty of dropping an easy catch offered by Travis Head, who eventually remained unbeaten.
England's spirits were lifted when Labuschagne was dismissed in bizarre fashion as his sweep ricocheted off Jos Buttler at short leg to Root fielding at midwicket.
Root's grab, though, garnered the ire of the departing Labuschagne with the third umpire ruling the replays to be not conclusive enough to overturn the on-field 'out' decision.
Later, fading light forced the umpires to intervene and disallow Archer from bowling leading to Root opting for part-timer Denly for the last few overs of the match.
Ben Stokes was adjudged man-of-the-match for his unbeaten century brought up in the first session of the fifth day at a time when the Australian pacers were probing for early breakthroughs.