Skipper Brathwaite, who took the match into a final day with a defiant overnight 101 after Australia declared their second innings at 182-2 on day four, couldn't keep the momentum going against Lyon and was bowled for 110.
The Caribbean team resumed on 192-3, needing to bat through 90 overs for a draw after gamely rebuffing an attack missing the injured Pat Cummins for two sessions on Saturday.
But they were all out for 333 after some plucky late resistance.
Lyon ended with 6-128, in the process surpassing India's Ravichandran Ashwin to become the eighth-highest wicket-taker in Test history with 446.
"The batters did their job, the quicks stood up, it was a fantastic all-round performance," said Australian skipper Pat Cummins.
"Lyon is the most important bowler in our team, he keeps taking wickets. There wasn't much on offer for the spinners, but he found out a way and kept picking wickets."
It extended the West Indies' 25-year run of failing to win a Test on Australian soil, with the hosts retaining the Frank Worrell Trophy ahead of the second, and final, Test in Adelaide starting Thursday.
"Obviously disappointed with the loss. We gave away the wickets in the first innings, but it was a good fight as a team and as a bowling unit," said Brathwaite.
"It's good to see the fight from the batters, but still a lot of improvement to be done."
Brathwaite said after his day four exploits that the opening overs would be "very, very crucial", and in front of a spartan crowd, it was Australia who prevailed.
Kyle Mayers, who has been ruled out of any further bowling in the two-Test series with a shoulder strain, added just 10 runs to his overnight nought.
Lyon made the breakthrough, with Mayers getting a thick edge to Steve Smith at slip.
Brathwaite was almost chanceless in making his 100, but Lyon, in his 111th Test, finally beat the bat and he was bowled.
"I was pretty happy with that one. Kraiggy batted extremely well, he's a superstar batsman and I had to be patient," said Lyon.
Part-time spinner Travis Head was brought on and in his first over accounted for Jason Holder for three, with Smith taking a spectacular slip catch.
When Australia took the new ball, Josh Hazlewood enticed an outside edge from Joshua Da Silva which Usman Khawaja collected at second slip to leave the West Indies in trouble.
But Rostan Chase and Alzarri Joseph (43) put up resistance, adding 82 for the eighth wicket before Head stepped up again to bowl Joseph.
Lyon took his fifth to remove Chase for a battling 55, then Kemar Roach for a duck to wrap it up.
- Double tons -
The West Indies never recovered from a dominant Australian first-innings batting display with Marnus Labuschagne and Smith both smacking double centuries to propel them to 598-4 declared.
After the West Indies were removed for 283 in reply, Labuschagne belted 104 not out to become only the eighth player to achieve the 200-100 feat in the same Test, joining the likes of Sunil Gavaskar, Brian Lara and Kumar Sangakkara.
While outplayed, a young West Indies did show grit.
Few expected them to take the game to five days against the world's top-ranked side, with their new opening partnership of Brathwaite and debutant Tagenarine Chanderpaul showing promise.
But their bowlers let them down, taking just six wickets for 780 runs.
Their stocks were, however, depleted with allrounder Raymon Reifer ruled out injured before a ball was bowled.
Top-order batsman Nkrumah Bonner suffered a concussion, Mayers was unable to bowl in the second innings with a shoulder strain and Roach limped off with a hamstring niggle.
With the injuries piling up, uncapped Marquino Mindley has been called up as cover for the Adelaide Test.