Chasing 123 for victory, the visitors crashed out in 13.4 overs with Shakib returning figures of 4-9 with his left-arm spin in the 60-run rout in Dhaka.
Australia's previous lowest was 79 all out against England in 2005.
Bangladesh, who had secured their first series win over Australia after winning the opening three matches, bounced back from their previous loss to flatten the opposition batting.
"The boys showed a lot of character and hunger to win a series against a team like Australia," home skipper Mahmudullah Riyad said after the triumph.
"We executed our plans really well. Obviously it was hard for the batters, but we did well to keep getting 120-130. All the bowlers worked really well together."
He said, "We always felt that in our backyard we are a very good team. We can't let anything go past.
"Though the ranking doesn't show it, I have always felt we have very good potential to be a very good team in T20 internationals."
Pace bowler Mohammad Saifuddin combined with Shakib to rattle Australia with twin strikes in one over and send back Alex Carey and Moises Henriques, both for three.
Shakib, who got Ashton Turner out for his 100th T20 scalp, wiped off the tail with the wickets of Nathan Ellis and Adam Zampa as Bangladesh celebrated.
The 34-year-old Shakib, a veteran of 58 Tests, 258 one-day matches and 84 T20s, said he is "still enjoying the game".
Spinner Nasum Ahmed had done the early damage when he bowled Dan Christian for three and then trapped the in-form Mitchell Marsh for four in successive overs.
Skipper Matthew Wade (22) and Ben McDermott fought back with a 21-run stand for the third wicket but Mahmudullah broke the partnership. He got McDermott caught and bowled for 17.
"I've played for a few years now, certainly the toughest conditions to play T20 cricket in," said a disappointed Wade. "Great learning for youngsters."
Earlier Mohammad Naim made 23 in Bangladesh's 122 for eight after electing to bat.
Christian, who took an international scalp after seven years, and Nathan Ellis took two wickets each.