Accies midfielder Darian MacKinnon volleyed in the opener in the 39th minute, but attacker Sean Winter levelled from the spot in the 69th minute.
However, in the 81st minute Louis Longridge flicked in a Grant Gillespie cross before Ali Crawford clinched the points for the Ladbrokes Premiership side with a close-range finish in the 90th minute.
Last season’s Lowland League champions had the best two chances of the first half, but striker Aidan Ferris missed them both, failing to hit the target in the first minute and then in the 20th minute when defensive lapses from the Ladbrokes Premiership outfit left him with only goalkeeper Gary Woods to beat.
However, home number one Matthew McGinley was by far the busier keeper, making a series of saves from the visitors, notably clawing away a Rakish Bingham lob and parrying a close-range header from Greg Docherty.
However, six minutes from the break McGinley had no chance when midfielder MacKinnon raced in to meet a Crawford corner and drove the ball high into the net from around 10 yards.
In the 55th minute, with the heavy rain doing little to help either side on the artificial surface, East Kilbride escaped when Bingham somehow managed to sky the ball over an open goal after McGinley had parried a Dougie Imrie shot straight to his feet.
At the other end minutes later, Adam Strachan crashed a free-kick from 25 yards off the crossbar with Woods beaten, but Kevin McCann could not turn the rebound into the net.
East Kilbride kept plugging away and levelled from the spot when Accies defender Georgios Sarris clumsily clambered over Craig Howie in the penalty area, with Winter sending Woods the wrong way with his perfectly-executed penalty.
However, with nine minutes remaining Longridge got on the end of Gillespie’s delivery from the left to flick the ball past McGinley with time left for Crawford to knock in a third.