The (Deceuninck-Quick Step) sprinter made the most of his pure speed to snatch the stage win in Madrid with Rogic (Jumbo-Visma) staying clear of trouble to become the first Slovenian to win one of cycling's three Grand Tours.
"It’s a great feeling to win this race, and to be accompanied by another Slovenian is even better," Roglic said. "It’s really nice for our cycling. I just tried to finish today again. We’re writing a bit of history.
"I’ll only know once I finish my career [if it’s the beginning before more successes]. But to win one Grand Tour is already a nice achievement."
Jakobsen out-sprinted the Irish national champion Sam Bennett (BORA-Hansgrohe) to claim his second Grand Tour victory with Szymon Sajnok (CCC Team) third.
“To be honest, I didn’t expect to arrive to Madrid," he said. "I suffered a lot in these three weeks, it was very hard, but I kept going and I’m glad to be here now.
"The first victory (Stage 4) brought a huge relief, because it took off the pressure just four days into the race, while today’s success was completely unexpected.
"The guys were again fantastic, controlling the break and bringing me to the front when it mattered. All I had to do was follow Max and launch my sprint at the right moment, and on the line, I even threw my bike just for good measure.”
Alejandro Valverde (Movistar) was second overall with Roglic's compatriot Tadej Pogacar (UAE Team Emirates) in third.
Miguel Angel Lopez (Astana) earned the most aggressive rider of the race which saw him finish fifth behind Nairo Quintana (Movistar).
Roglic, who had lined up as one of the favourites to succeed absent defending champion Simon Yates (Mitchelton-Scott), seized control of the overall standings after dominating his rivals on the Stage 10 individual time-trial then proved unshakeable through to the final stage.
But he survived a few scares along the way but lined up for the traditional last day general classification procession with a lead of over two and a half minutes from Valverde.
While Roglic was the overall winner, his 20-year-old compatriot Pogacar not only climbed the podium but also secured the white jersey as best young rider.
The UAE Emirates Team rider was the breakout star of this year's Vuelta, and deprived Lopez of the white jersey.