A game low on quality was heading for a goalless conclusion when Hendrick tapped home Johann Berg Gudmundsson’s fine cross after Rob Elliot had beaten away Jack Cork’s effort.
Newcastle had earlier gone close when Ayoze Perez’s curling attempt from the edge of the penalty area forced Nick Pope into a full-stretch diving save.
Joselu drilled another chance wide in the 83rd minute as the visitors fell to a 12th defeat in their last 16 top-flight away games.
Dyche has been linked with the manager’s job at Everton.
“I just keep going on with my job,” he said, when asked about the vacancy at Goodison Park after the game.
When Dyche took charge of Burnley on 30 October 2012, the Clarets were 14th in the Championship table.
Now they are seventh in the Premier League – level on points with sixth-placed Liverpool – after their 85th win in 211 league game under the former Watford boss.
“It’s a nice looking league table but it’s not a sprint. It’s a marathon and we have made a good start,” added Dyche.
“We never take anything for granted here because you can’t do in this division as it’s very tough.”
This is Burnley’s third season – and second in a row – in the top-flight under Dyche.
No longer are the Clarets seen as cannon fodder at this level.
This season, they’ve won at Chelsea and drawn away to both Tottenham and Liverpool, while they are now unbeaten in the past four at Turf Moor.
They have played better than they performed against Newcastle in a match which only came to life in the closing quarter.
The Clarets were missing club record £15m signing Chris Wood due to a hamstring injury and it looked like they were going to have to settle for a point until Hendrick pounced for his second goal in his last four league games.
The Republic of Ireland international could not miss after Gudmundsson’s quality cross before, late on, Elliot denied Ashley Barnes from doubling the lead with a one-handed stop.
Newcastle seemed happy to settle for a point and only showed any real urgency once they fell behind after Perez had carelessly lost possession.
“We had the game under control,” said Magpies boss Rafael Benitez. “I’m really disappointed.”
The visitors twice went close to equalising in the closing moments – Pope keeping out substitute Isaac Hayden, before Javier Manquillo’s flicked header from Jonjo Shelvey’s cross landed on the roof of the net.
On another Monday night to forget for the Magpies, Christian Atsu was guilty of wasting the best chance of the first half when he poked over from close range.
Newcastle, who have failed to score in four of their 10 league games this season, have now lost 10 consecutive top-flight matches on a Monday.
Asked what his side is missing, Benitez added: “Be more precise on the ball and be more clinical. We made two or three crosses that were not the best when we were in a very good position.