Amabhubesi scored through defender Ahmed Mohammed and striker Bernard Parker, while Pirates pulled one back late on through lively winger Bennett Chenene.
Pirates' Francis Chansa was certainly the busier of the two goalkeepers in the first half, making some good saves to keep his team on level pegging with the hosts.
In the sixth minute the DR Congo keeper palmed away Ryan Botha's header and a quarter of an hour later he tipped Parker's 20-yard drive around the post for a corner kick.
Just past the half-hour mark Pirates threw away a good chance to break the deadlock, with Paulus Masehe shooting wide from close range after Mandlenkosi Tswenga had parried Teko Modise's free-kick.
The match was goalless at halftime.
Thanda went a goal up in the 49th minute, as Mohammed got on the end of a set-piece and headed the opener.
The hosts almost added a second just over 10 minutes later when Chenene had to clear Botha's corner off his own goal line.
In the 77th minute Amabhubesi did manage to double their advantage, as Parker got on the end of another fine corner-kick from Botha and headed his team into a 2-0 lead.
It didn't last long though, as Chenene scored with a low shot some three minutes after Parker's goal to halve the deficit.
Thanda remain in 14th place, but they are now six and nine points clear of Jomo Cosmos and Black Leopards respectively. Pirates remain in third on 35 points but their hopes of winning the title are fading fast.