Derek McInnes' side trailed to Dujon Sterling's deflected effort in the first half at Tynecastle.
But Hearts regrouped at the break and Stephen Kingsley equalised before Shankland's winner sparked wild celebrations.
The Jambos, who have won their last three games, are now three points ahead of second-placed Celtic with a superior goal difference.
They are seven points clear of third-placed Rangers, with just three games remaining in a gripping title race.
"The way the players dug that out was magnificent," McInnes said. "It was a change of approach, a change of mindset, we had it all to do.
"We needed a big half-time, and thankfully we organised ourselves enough to get more confidence in the game."
Hearts are hoping to become the first club other than Celtic and Rangers to win the Scottish crown since Aberdeen in 1985.
They face Celtic in a potential shoot-out for the title on the last day of the season, but could wrap up the title before then.
"We asked the players to show more responsibility," McInnes said.
"We needed to start showing we were the team at the top of the league because we were miles off it at the start."
Rangers boss Danny Rohl added: "To win such a game, it is not enough to play for 45 minutes on a high level, you have to play 90 minutes on your highest level.
"I think if we're honest with each other in the last three games, we have not always had enough to take three wins."
McInnes said on Friday he wanted to see Tynecastle "in all its glory" for one of its biggest-ever games.
He got his wish as his team emerged into a sea of maroon and white and a cacophony of noise.
Rangers spurned a golden chance in the 19th minute when Tochi Chukwuani sent a free header over the bar from a James Tavernier corner.
Rohl's men silenced the home support in the 23rd minute as Tavernier's long throw was cleared only as far as Sterling, whose volley from eight yards out deflected in off Michael Steinwender.
McInnes responded by sending on Blair Spittal for the start of the second half and the change helped spark Hearts.
Shankland saw a ferocious shot blocked by Emmanuel Fernandes before Claudio Braga forced Jack Butland into his first save.
The equaliser came in the 54th minute when Alexandros Kyziridis cut in from the right and curled a low shot off the far post, with Kingsley perfectly placed to guide in the rebound from just outside the six-yard box.
Hearts grabbed the decisive goal in the 71st minute when Spittal released Kingsley down the left and his cutback took a deflection off Tavernier.
It fell kindly to Shankland, who dispatched an emphatic low finish beyond Butland as Tynecastle erupted.