Bulli, 22, who was the competition's leading wicket-taker last season, claimed 4-50 and 5-22 with his left-arm unorthodox spin as Kensington, after making 303 in their first innings, bowled out Boys' Town for 125 twice in their clash at Kensington Park.
Ryan Francis, 55, had earlier supported Bulli to set up Kensington's challenging total.
Top of standings
The victory propelled Kensing-ton to the top of the standings along with the Jamaica Defence Force (JDF) which, on a brilliant all-round performance from Joseph Palmer, also defeated Police outright in their game at Elletson Road.
Palmer first made 63 in the JDF's first innings total of 351, before returning to claim 4-38 and 6-46 to restrict Police for 154 and 144, respectively. Avroy Harris, 84, made most runs when the soldiers batted, when Rensford Boadie took 7-127.
AT EDEN PARK: St Mary 187 (Orane Farr 59 not out; Romeo Dunka 5-50) played to a no-result against GC Foster 62-2 (rain) (Orville Pennant 62, Phillip Pinnock 30; Orane Farr 1-9).
AT FOLLY OVAL: St Thomas 164-3 (rain) (Jamie Trenchfield 70 not out, Oniel Wright 44 not out). No-result against Portland.
AT NELSON OVAL: Kingston and St Andrew 246 (Shane Harris 48, Damion Foster 37; Ransford Merchant 4-72) and 131-9 (Keron Edwards 35; Ricardo Ennis 5-57) lost first-innings points to Lucas, 257-8 (Ramon Senior 97, Wayne Gayle 38; Shane Harris 4-49) and 55-2 (Kevin Hinds 22; Renor Ross 1-15).