Australia had posted a big total of 193/6 thanks to Jake Fraser-McGurk's first international 50 and contributions from captain Travis Head (31) and Josh Inglis (42).
But Livingstone had a starring role with both bat and bowl to set up a winner-takes-all clash for the series in Manchester on Sunday.
In his 50th T20 international, the spinner was the pick of the England bowlers with 2-16 despite only bowling three overs.
He then came to the crease with the hosts 34-2 inside four overs.
Stand-in captain Phil Salt's 39 off 23 deliveries helped his side rebuild but it was a blistering 90-run partnership off 47 balls between Livingstone and Jacob Bethell that swung the game England's way.
Bethell departed for 44 on just his second international but Livingstone remained at the crease until the scores were tied.
Livingstone was denied hitting the winning runs when he was bowled by Matt Short, who took 5-22, but Adil Rashid's single saw England over the line with six balls to spare.
Earlier, England looked set for a chase in excess of 200 as the tourists made another rapid start with Head smashing his 31 off just 14 balls.
Australia were cruising along at 119-2 in the 13th over before Fraser-McGurk's flashy innings came to an end when he was caught off the bowling of Livingstone.
Just like in the opening game of the series, which Australia won by 28 runs, it was left to the English spin attack to limit the damage.
But Inglis' 42 off 26 balls put momentum back into the Aussie innings before Cameron Green and Aaron Hardie hit Sam Curran for 20 off the final over to post a total of 193/6.
