The impressive Dimitar Berbatov had given Fulham a first-half lead, but the muscular presence of Romelu Lukaku drew the hosts level early in the second period.
However, Kacaniklic rolled home the decisive goal just before the hour mark to hand Fulham only a second win in 13 games to lift Martin Jol's side away from the relegation places.
A New Year hangover seemed to be affecting both teams for much of the first half, with a lack of fluency to the game and neither goalkeeper troubled.
Gareth McAuley and Gabriel Tamas headed set-pieces wide for West Brom, while at the opposite end Berbatov and Giorgos Karagounis both fired over from outside the penalty area.
A rare moment of class produced Fulham's opener six minutes before half-time as Berbatov sent Ashkan Dejagah scampering down the right before the Bulgarian ambled into the box to sweep home his team-mate's return pass.
After a lacklustre first half, West Brom were a different team in the second period and were level within four minutes when Lukaku was left alone inside the six-yard box to turn home Billy Jones' low pass following clever work by the full-back.
Lukaku, who has impressed during his loan spell from Chelsea, almost gave West Brom the lead three minutes later when his thumping volley came back off the upright.
The Belgium international was in the thick of things again on 55 minutes as his shot was kept out by David Stockdale before James Morrison struck the rebound against the post and Chris Brunt saw his goalbound effort diverted clear by Aaron Hughes.
Yet the Baggies were floored by a sucker-punch on 58 minutes as the recalled Kacaniklic superbly controlled Bryan Ruiz's pass on the edge of the box before slotting a low shot past the stranded Ben Foster.
Steve Clarke's side took time to recover from that setback and turned to Peter Odemwingie and Shane Long for inspiration, with the latter's cross 15 minutes from time headed wide by Lukaku.
Fulham were still a threat on the counter and could have had a penalty with 10 minutes to go when Ruiz was caught by Youssouf Mulumbu, but it mattered little as only a late Jones cross threatened to deny Jol a win against the club he represented during his playing career.