Sharks and Stormers claim first ever Champions Cup last-eight spots
Mbonambi's tries against the two-time former winners came in a four-minute spell in the second-half to set up a last-eight tie next weekend with either Toulouse or the Bulls, who meet on Sunday.
The Durban-based outfit are one of three South African sides featuring in the Champions Cup this season for the first time, along with the Bulls and the Stormers, whose flanker Deon Fourie also claimed a double in the victory over Quins.
"It was a tough outing, it was really hot out there," Sharks and Boks captain Siya Kolisi told SuperSport.
"We knew what we had to do, they're a great side, they have a great history in this competition," he added after the win in 26 degrees sunshine on the Indian Ocean coast.
Mbonambi's Test team-mate lock RG Snyman made his first Munster start since August 2020 after recovering from two serious knee injuries.
"It was good to see RG back," Kolisi said. "I know a lot of South Africans are happy with that."
The Sharks, who are partly owned by an American consortium in partnership with rapper Jay-Z's Roc Nation agency, led 17-14 at the break before Mbonambi's first double since the successful 2019 Rugby World Cup campaign.
Both of the 32-year-old's tries came from short range with the second off a dominant rolling maul with a little over half an hour to play.
Mbonmabi's fellow Boks, scrum-half Jaden Hendrikse, lock Eben Etzebeth, winger Makazole Mapimpi and fly-half Curwin Bosch as well as Rio Olympic Games bronze medallist Werner Kok also went over for the home side.
Willemse acrobatics
In similar heat in Cape Town, 36-year-old Fourie set the Stormers up for victoryunster by crossing after just 55 seconds and added a second just before the half-hour mark.
Springboks prop Steven Kitshoff, full-back Damian Willemse, with an acrobatic touchline finish, and replacement flanker Willie Engelbrecht also scored for the side from the Western Province.
England No. 8 Alex Dombrandt with two, former Sharks centre Andre Esterhuizen and winger Joe Marchant claimed the Quins' tries in front of 40,000 people.
The Stormers' reward will be a fixture against either 2020 winners Exeter or Top 14 champions Montpellier, who play in England on Sunday.