Llanelli went in 10-3 down at the break after a Munster try for Sean Henry and conversion and penalty for Declan Cusack to just one Shingler penalty, but the visitors fronted up to a tough physical battle and eventually came out on top, with Shingler's boot on target all evening.
Both sides were down to 14 men at the end of the first half with Aaron Shingler then Peter O'Mahony going to the bin.
Munster missed a penalty in the dying moments but Llanelli's efforts throughout the 80 minutes meant they deserved a stunning victory.
"This win is up there with the best moments I've ever experienced, including at the very top level of European rugby," said Llanelli's head of rugby Anthony Buchanan.
"To come here and beat a very experienced Munster A side is hugely pleasing, especially given that we had to dominate a forwards battle in order to come away with the spoils.
"The victory speaks volumes for the collective attitude and team spirit and I would hope we can take a lot from this and progress further.
"No8 Ben Morgan was once again outstanding, and fellow back row Nic Cudd certainly stepped up to the plate. However, the rest of the team were also impressive. We lost a few players to injury but to a man, the players showed commitment and courage."
Elsewhere in the British & Irish Cup, Pontypridd won 28-18 at Llandovery thanks to 13 points from the boot of Simon Humberstone and three tries to the hosts' two.
Swansea's unbeaten start to the season was ended by Ulster Ravens at St Helen's, with the Ulstermen running out 32-12 winners.
In the only other match of the rescheduled Round 5 fixtures, Newport lost 33-28 in a thriller at home to Leinster A.