Craig Halkett's stoppage-time winner sent Hearts five points clear of Celtic at the top of the Scottish Premiership after a 1-0 win over Edinburgh rivals Hibernian on Saturday.
Hearts are aiming to end a 40-year wait for anyone other than Celtic or Rangers to win the Scottish title.
The Jambos have been boosted by a major investment by Brighton owner Tony Bloom, who bought a 29 per cent stake in the club in June.
Hearts have dropped just two points from their opening seven league games under Derek McInnes to spark hope of a title challenge.
There were euphoric scenes at Tynecastle after Halkett forced in a cross from substitute Sabah Kerjota to secure a first victory over Hibernian in five attempts.
"It's great we found a way to win. We're only seven league games in, but we've had three last-minute winners which says a lot about the team, the squad depth, the mentality," said McInnes.
"We would like it to be more comfortable, but the rivalry is so fierce and it's very rare these games are tight."
The visitors had missed the best chances through captain Martin Boyle, who also had a goal ruled out for offside.
Defeat leaves Hibs down in seventh, 11 points behind their city rivals.
"It sums up our season so far, not taking our chances when they come and not seeing out the game at the end," said Hibs boss David Gray.
"Our points return at this stage isn't good enough. We need to start climbing this table quickly and getting some wins."
Celtic can close the gap at the top back to two points when they host Motherwell on Sunday.
Eighth-placed Rangers, who are now 12 points adrift of the leaders, are aiming to build on their first league win under Russell Martin last weekend when they travel to Falkirk.