Rus won through 6-0, 7-6 in one hour and 45 minutes to deny wild card Akugue a dream victory in her first WTA main draw event.
In the men's tournament, Germany's Alexander Zverev (26) crushed Frenchman Arthur Fils (19) to qualify for his first final in over a year, where he takes on Serbian Laslo Djere (28) on Sunday.
"I'm very proud of myself that after so many years, I made my first title," said Rus, who turned pro back in 2008.
She was playing in her 126th tour-level main draw this week, making her the oldest first-time title-winner in 40 years.
"The day was very long because I was very nervous, but I'm happy that I could play good tennis and it was a great final from both of us."
In a match between two left-handed players, seventh seed Rus won the opening set without the loss of a game.
Akugue, born in Hamburg to Nigerian parents, fought back in the second with an immediate break, but Rus responded straight away with both holding serve to force a tie-break.
Akugue saved two championship points to hold serve in the 10th game of the second set, and two more in the tiebreak, before Rus won through.
Former world junior one Rus had failed to live up to her early promise after winning the girls' singles title at the Australian Open in 2008.
She came into Hamburg at a career-high 60th ranking, and will break into the top 50 for the first time on Monday.
Zverev ends long wait
Zverev eased past Fils 6-2, 6-4, the 19th-ranked German storming to a 4-0 lead to wrap up the first set in 37 minutes.
In the second set, after a brief rain delay, Zverev again used his serve and numerous unforced errors from his 71st-ranked rival to seal victory in one hour and 31 minutes.
It will be the German's first final since Madrid in May 2022 having suffered a serious ankle injury at the French Open last year.
In Sunday's final, he will face 57th-ranked Djere who eased past China's Zhang Zhizhen (26) 6-3, 6-2.
The 26-year-old Zverev is chasing his first title since his triumph at the 2021 ATP Finals, and 20th of his career.
Djere, 28, is bidding for his third ATP title after Rio de Janeiro in 2019 and Sardinia in 2020.
