Mbonambi's contributions came in a four-minute spell in the second-half to set-up next weekend's last-eight tie 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, who welcome Harlequins later on Saturday.
Mbonambi's Test team-mate lock RG Snyman made his first Munster start since August 2020 after recovering from two serious knee injuries.
The visitors, two-time former champions, led 7-3 after 10 minutes as winger Shane Daly crossed in response to fly-half Curwin Bosch's early penalty.
The South Africans then responded to reclaim the lead with two tries in the space of 10 minutes.
Scrum-half Jaden Hendriks dashed to score after a brilliant Makazole Mapimpi break and Snyman's fellow Springboks lock Eben Etzebeth crashed over from short range.
Bosch's two conversions made it 17-7 before Munster prop Dave Kilcoyne scored his first try since September 2018 from a pick and go.
Jack Crowley's additional points in 26 degrees sunshine on the Indian Ocean coast made it 17-14 at the break.
The Sharks claimed their last-eight spot with Mbonambi's two efforts within the opening eight minutes of the second half.
Both tries came from short range and the second from a dominant rolling maul for his first double since the 2019 Rugby World Cup.
Winger Werner Kok, playmaker Bosch and Mapimpi also crossed for the home side to make sure of a place in the next round.
Replacement hooker Diarmuid Barron, full-back Mike Haley and substitute lock Fineen Wycherley claimed consolation efforts for the 2006 and 2008 winners.
Later, holders La Rochelle host Gloucester and four-time winners Leinster face Ulster in an Irish derby in Dublin, with the victors hosting Leicester in the next round after they beat Edinburgh 16-6 on Friday.
On Sunday, 2020 champions Exeter play Montpellier and Saracens host the Ospreys.
