The pair scrapped all the way to the finish of the 135km final stage around Barcelona including multiple ascents of the Montjuic mountain.
Evenepoel, 23, went into the seventh and final stage 10 seconds adrift of Roglic.
Hard as he tried, the Belgian was unable to shake off his rival, with Roglic, 33, allowing him the minor honour of taking the stage win without a sprint for the line, cutting the winning deficit to just six seconds.
"We expected his attacks. The seven days were difficult but I had the legs," said Roglic after winning the Tour for the first time.
"I'm happy to add my name to the list of winners of this historic race."
The Slovenian, who had already won the previous week's Tirreno-Adriatico in Italy, took the lead in the general classification by winning the first stage.
The two leaders did not let go of each other on the steep roads of the Catalan hinterland, with the Belgian winning the third stage ahead of Roglic.
The Slovenian, though, set up the overall victory by dropping Evenepoel on the upper slopes of the final climb of Friday's stage.
"I knew it was going to be difficult but I had to try. It was a good battle. I'm very happy with my week," said Evenepoel.
Joao Almeida of UAE Team Emirates finished third overall, over two minutes behind Roglic.
Roglic lost out to Evenepoel at the Tour of Spain last season, having won the previous three editions, and the pair were using Catalunya as a warm-up for their big showdown at the Giro d'Italia in May.
