Their first home tie in the tournament for 26 years ended in a comprehensive 3-0 win, and it was Konta who was responsible for the hard yards.
She eventually beat Dalila Jakupovic 7-6 6-2, saving three set points in the first set, to give Great Britain an unassailable 2-0 lead.
They had gone 1-0 ahead after Katie Boulter won 6-4 6-2 against Kaja Juvan. Harriet Dart and Katie Swan later won the doubles rubber to put the gloss on the victory.
Captain Anne Keothavong was delighted to get the job done easily ahead of what will be a gruelling few days.
“I didn’t really want to be sitting there in a deciding doubles on day one!” she said.
“Credit to Boulter and Konta. It is a much nicer situation knowing we have won, but this competition is far from completed.
“Physically it’s tough, it’s much warmer, but mentally you can’t under-estimate all the emotions they go through, playing in front of a home crowd for the first time.
“I thought Katie Boulter was brilliant, making her singles debut and winning in front of her home crowd is special.
“She has improved so much in the last 12 months and I think 2019 can be really special.”
Great Britain next play Greece on Thursday as they aim to reach the the World Group II for a third year in a row.
Croatia, Turkey, Serbia and Georgia are in the other pool. Croatia got their campaign off to a winning start with a 2-1 victory over Turkey, while Serbia beat Georgia by the same scoreline.
The winners of each group will play each other to determine who advances to the play-offs in April.