At a glance
The dominant Edinburgh side surged eight points clear of their Glasgow rivals at the top of the Scottish top flight to strengthen their early championship claims and worsen the struggles around Brendan Rodgers' side.
Alexandros Kyziridis' fine strike and Lawrence Shankland's penalty gave the Hearts team a 3-1 lead that the reigning champions could not overcome.
Dane Murray - one of three youngsters in the Celtic starting lineup - slashed into his own net in the early stages to put the home side ahead at a energetic their home ground, and it was the 22-year-old's foul that led to the spot-kick.
A stylish finish from McGregor of the season had swiftly equalized Hearts' opener, and Benjamin Nygren almost put Celtic in front, choosing placement over power and allowing Alexander Schwolow to save from close range.
But the unstoppable Hearts side seized control again to hand Celtic to back-to-back losses for the initial occasion since 2023.
It also gives the Hearts a commanding advantage at the summit.
The previous occasion they were this far ahead was in the 2005-2006 season under George Burley, when they ended as runners-up - one of just four occasions that a team has not secured the title after building such an advantage in the last three decades.
McInnes, whose former team were second to Celtic on multiple times, has been working hard to downplay championship discussions, as the club seek their first top-flight crown since 1960.
The Celtic boss has experienced this previously at Celtic - both perfect and shaky beginnings to a season. He's never ended as runner-up.
But on Sunday's evidence, this Hearts team showed they deserve serious consideration.
Hearts started with the self-assurance of a side unbeaten in the league this season - fans' favourite Kyziridis delivered a ball for Shankland to fire off target.
Celtic didn't help themselves with some unusual sloppiness in possession and they buckled under their first real examination.
Claudio Braga's header was touched on to a post by the Celtic keeper and the follow-up from Shankland was diverted goalwards by the hapless defender, in for the injured Cameron Carter-Vickers.
The away side found their rhythm and started stringing passes together. From one such move, Sebastian Tounekti released the full-back to square for the captain, who rolled in.
Aside from the opportunity for Nygren, Celtic struggled to create much for lone frontman Jonny Kenny or his replacement Shin Yamada amid the unavailability of stricken attackers Kelechi Iheanacho and Daizen Maeda.
The home team regained their intensity and versatility in attacking from across the final third. Kyziridis, a standout performer of the club's relationship with shareholder Tony Bloom's Jamestown Analytics, fired a low shot past the goalkeeper from the edge of the box.
Immediately after, the defender fouled the midfielder in the inside right channel and Shankland converted. And it could've been an even more handsome win, had substitute Ageu not been stopped by Schmeichel.
The visitors' performance after the break was partially encapsulated in two incidents involving record signing Arne Engles.
The set-piece from Engles delivery was in stark contrast to his midweek exploits in the continental victory over Sturm Graz and the Belgium international was then yellow-carded for a professional foul.
Hearts head coach Derek McInnes: "The effort from the squad is evident, the team spirit. That's not always enough to win games consistently.
"Our strength lies in, is good technical ability. You're witnessing it from multiple individuals who have been here previously. Recent additions are eager to show that.
"We're in a positive moment at the minute, but it's only three points. It's crucial for us to maintain this form. We've got loads to do, loads to do. After the first two rounds is typically a benchmark of where you are."
The Celtic boss: "The second goal was disappointing from our perspective. We failed to push up when the ball went back as a defensive unit so when they pass the ball, they should have players offside but didn't. We have got to do better with the attempt.
"Plenty of matches to play for. It's still so early. It's just nine matches. Clearly, being eight adrift is not ideal. The positive is there's numerous fixtures to play.
"We now have to focus on the present and look to find more consistency and better performance in our play."
Celtic are returning to their stadium on Wednesday against newly-promoted Falkirk (evening kick-off). At the same time in the Scottish Premiership, Hearts are at their opponents' ground.
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