England, set a daunting 398 for victory, lost four wickets for 12 runs either side of lunch on the last day as they collapsed to 97-7.
They were eventually dismissed for 146 in their second innings, with off-spinner Lyon taking 6-49 -- a haul that included his 350th Test wicket -- and fast bowler Pat Cummins 4-32.
Victory gave Australia their first win at Edgbaston in any format since 2001 -- the last time they won an Ashes series in England -- and ended England's run off 11 successive wins at the Birmingham ground.
England resumed on 13 without loss after Australia star Steve Smith's second hundred of his comeback Test following a ban for his role in a ball-tampering scandal had allowed the tourists to declare late on Sunday's fourth day.
Rory Burns, whose first-innings 133 was his first century at this level, became just the 10th cricketer to have batted on all five days of a Test.
But he had added just four runs to his overnight seven when he was undone by an excellent seaming and rising Cummins delivery that lobbed off his gloves to Lyon in the gully.
Number three Root, on four, was given out leg before to James Pattinson by umpire Joel Wilson.
But in a match full of overturned decisions, Root successfully challenged the verdict.
Root had made eight when he nearly played on to Cummins before Jason Roy, who made a superb 85 in eventual champions England's World Cup semi-final win over Australia at Edgbaston last month, drove him through extra-cover for four.
Root had moved on to nine when he was again given out leg before by Wilson, this time off the bowling of paceman Peter Siddle. But the batsman's immediate review revealed an inside edge.
That meant umpire Wilson had equalled an unwanted record with an eighth overturned decision in a Test match.
Roy swept Lyon for four but then, in a shot more appropriate for a one-day match than a side trying to save a Test, charged down the pitch and was clean bowled for 28
Lyon remained a threat on a wearing pitch offering turn and bounce and he had Joe Denly (11) caught off bat and pad by Cameron Bancroft at short leg to leave England 80-3.
And the Australia pair combined again to dismiss Root for 28.
England's 85-4 at lunch soon became 85-5 when Jos Buttler (one) was bowled by a Cummins delivery that kept low.
Cummins then had his 100th Test wicket when Jonny Bairstow (six) gloved a rising ball to Bancroft in the slips.
England's collapse continued apace when Ben Stokes was also out for six, well caught by wicketkeeper Tim Paine off a sharply spun Lyon delivery.
Moeen Ali, who in the first innings had succumbed to Lyon when he was bowled for a duck playing no shot, fell to the bowler for the ninth time in 11 Ashes knocks when caught in the slips for four to leave England 136-8.
It marked the end of a miserable match for England off-spinner Moeen, comprehensively outbowled by Lyon.
Lyon had Stuart Broad caught in the slips next ball before James Anderson survived the hat-trick after not bowling at all in Australia's second innings because of a calf injury.
Chris Woakes, who played several fine shots, was dropped in the slips by Smith.
But former Australia captain Smith had the last word, catching Woakes (37) in the slips off Cummins, with England all out inside 53 overs.