The 29-year-old off-spinner completed a career-best match haul of 13-154 to help bundle out the hosts for 157 on day four at Chittagong's Zahur Ahmed Chowdhury Stadium.
Chasing 86 for victory, Australia suffered a mini-collapse of their own, losing openers David Warner and Matt Renshaw as well as captain Steve Smith before reaching the 50-mark.
Glenn Maxwell made a breezy 25 not out, though, and sealed the victory with a six off Nasir Hossain to avenge Australia's 20-run loss in the series opener in Dhaka.
"I thought we played some good cricket at times in this Test match," Smith said at the presentation ceremony. "I'm proud of the way the boys got over the line here in the end to get the 1-1 draw."
Earlier, Australia were not able to add to their overnight score as Lyon fell for a duck and were all out for 377, a first innings lead of 72.
Paceman Pat Cummins drew first blood for the tourists by dismissing opener Soumya Sarkar before Lyon's three-wicket burst in the morning session unhinged the home side.
By the time they had erased the first innings deficit, Bangladesh had lost the top half of their batting line-up and the lack of a partnership down the order effectively condemned them to defeat.
Captain Mushfiqur Rahim top-scored with 31, while Mominul Haque (29) and Sabbir Rahman (24) also got the starts but none of them were able to move on to the sort of big knock that might have set Australia a bigger Test in the fourth innings.
Lyon claimed a third successive six-wicket haul to take his series tally to 22 wickets, the most by an Australian bowler in a two-Test series.
"I thought Nathan Lyon was remarkable in that first innings. There wasn't much in the pitch for him, and I think he got 22 wickets for the series, really good effort," Smith said of his spin spearhead.
Lyon won the man-of-the-match award and shared the man-of-the-series honour with teammate Warner who scored back-to-back centuries in the series.
Despite the draw, Australia slipped to fifth place in the official Test rankings.
Home captain Mushfiqur said his team should have batted better in the first innings when they scored 305. "Credit goes to Australia. We knew they are a very strong team and will come at us harder," said the stumper-batsman. "If we could score 350-400, a target of 150-200 in the fourth innings, you never know. Our first innings cost us the game."
COMMENTS
Comments are moderated and generally will be posted if they are on-topic and not abusive.
For more information, please see our Comments FAQ