"I think it is a bit of relief, and enjoyment," Dyche, whose side had lost their last three fixtures at Turf Moor while conceding nine goals, told the BBC.
"It was a hard-earned and hard-fought win. We never thought it would come from being free-flowing. We thought we would have to dig in and we have done that today."
Though the goal was credited to Tarkowski, it was Jack Cork who was responsible for the 40th-minute breakthrough when his shot deflected off the England defender's midriff into the net.
It was a scruffy goal but Dyche was not complaining.
"A goal is a goal. When you find wins and clean sheets hard to come by, 1-0 is a very pleasing result," he said.
"Performances have been suggesting we were getting back to that hardy edge we know we have got. We created chances but it wasn't about that. It was about fight and spirit. We know we can get better but sometimes you've got to grind."
The goal was only what Burnley deserved, though, after they took control in the second part of the opening period with Chris Wood and Robbie Brady, who had also volleyed narrowly over, forcing fine saves from Mat Ryan.
Burnley also felt they should have earned a second-half penalty when Yves Bissouma's extravagant bicycle kick in his own box caught Phil Bardsley.
Yet on a special day for Burnley's Joe Hart, who was making his 400th league appearance in English football, the former England goalkeeper proved his quality as the home side held on under increasingly severe pressure.
Brighton's manager Chris Hughton, whose side lie 11th in the table, felt the Seagulls deserved more.
"I thought we were the better side. I thought we had the better chances in the game," he said.
(Reporting by Ian Chadband; editing by Ken Ferris and Ed Osmond)
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