Celtic failed to cut the gap on the Scottish Premiership leaders, Hearts, after enduring their first defeat away against Dundee since 1988.

Celtic fans had disrupted the game early on by throwing dozens of balls on to the pitch in their ongoing protest against the club’s board, before Clark Robertson headed the home side ahead from a corner in the 17th minute. The Celtic defender Cameron Carter-Vickers then scored an own goal just before the break.

The visitors dominated the second half but lost 2-0, their first Premiership defeat of the season and first at Dens Park since Tommy Coyne scored the only goal of the game for the Dark Blues 37 years ago.

Celtic, booed at the end, remain five points behind Hearts, whom they play at Tynecastle next Sunday. Before then they have a Europa League tie against Sturm Graz on Thursday, with improvement required.

The game came to a quick halt after the first whistle blew when Celtic fans bombarded the pitch with different coloured balls, causing a delay while they were cleared. Visiting fans then chanted against the chief executive, Michael Nicholson, and the chair, Peter Lawwell.

When the game started it was low-key until the 10th minute when the home striker Joe Westley mis-kicked the ball six yards from goal after Simon Murray had cleverly stepped over a Drey Wright cross.

At the other end, Kelechi Iheanacho clipped the outside of the post with a snapshot but it was Dundee who forged ahead when the unmarked Robertson leapt to bullet in a header from a Cameron Congreve corner.

Clark Robertson watches his header fly towards the far corner to give Dundee the lead. Photograph: David Young/Action Plus/Shutterstock

Kieran Tierney and Yang Hyun-jun both missed the target with efforts as Celtic piled on the pressure. However, in another dangerous Dundee breakaway Congreve turned the Celtic centre-back Liam Scales before crossing for Westley to skim the ball in off the helpless Carter-Vickers and double the home side’s lead.

James Forrest and Johnny Kenny replaced Yang and Anthony Ralston for the start of the second half. Celtic kept up the pressure while Dundee defended resolutely, threatening any time they broke forward.

In the 58th minute Celtic’s Reo Hatate fired wide from the edge of the box before Ashley Hay, on as a Dundee substitute for Murray, came close with a header from 12 yards.

The Dundee goalkeeper Jon McCracken twice thwarted Iheanacho and it looked like Celtic were handed a lifeline when Hatate’s shot appeared to be blocked by the arm of Paul Digby inside the Dundee box. The referee, Matthew MacDermid, pointed to the spot but after he checked his pitchside monitor, at the behest of his VAR, he overturned his decision to the cheers of the home fans.

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The Celtic manager, Brendan Rodgers, who had previously expressed dissatisfaction with the way the summer transfer window transpired after Adam Idah and Nicolas Kühn left the club, said: “Listen, it’s not all linear and all smooth right the way through the season.

Celtic fans threw dozens of tennis balls on to the pitch in protest at the club’s board. Photograph: Stuart Wallace/Shutterstock

“I think the challenge from the summer, now leading into here, where we lost a lot of firepower, a lot of goals out on the team. And there’s no way you’ll go into a race and be given the keys to a Honda Civic and say: ‘I want you to drive it like a Ferrari.’ It’s not going to happen. So until something changes, I have to find the solutions. Because like I said, goals, speed, everything has come out of the team and we need to find a way to be better.

“It was clear in the summer, but it’s really the past now. There’s nothing we can do about it. We had the opportunities to do what we needed to do. It didn’t happen, so now it’s finding ways, whether it’s 4-3-3, whether it’s 3-4-3, whether it’s 3-5-2. We’re trying to look at all these different permutations within the team.”

The Dundee manager, Steven Pressley, said: “I’m exceptionally proud of the group of players. I think to beat Celtic on any given day, you have to carry a degree of luck. We also have to show a real resilience and a real determination in the way you play.”