By Mark Buckingham
Last Updated: 01/01/70 1:00am
Alex McLeish's men were forced to come from behind to secure the points with Portuguese attacker Capucho grabbing the decisive goal on 86 minutes after Gary McDonald and Peter Canero had earlier put the hosts in front.
The Gers broke the deadlock on the half hour when Shota Arveladze hammered the ball into the top corner after Peter Lovenkrands failed to connect with Mikel Arteta's corner.
Killie were level before half time after a slick move involving Steve Fulton and Canero presented McDonald with the opportunity to score at the far post.
The hosts were ahead five minutes after the break when Gary McSwegan's header dropped into the path of Canero, who steered the ball into the back of the net.
That setback sparked Rangers back into life with Emerson unleashing a powerful shot just wide and Michael Mols crashing an effort off the underside of the bar.
Arveladze should have drawn The Gers back on level terms but somehow missed the target after Francois Dubourdeau could only parry Emerson's shot.
On 63 minutes, Arveladze made amends with the Georgian striker finishing emphatically after a clever touch by Mols.
Lovenkrands then also hit the woodwork while at the opposite end, Arteta had to scramble the ball off the line from Martin Hardie's shot.
The match was settled four minutes from time when Arteta's free kick was headed home by Capucho, who may finally have endeared himself to the Rangers faithful after a lukewarm start to his Ibrox career.
Aberdeen's season lurches from bad to worse as they went down to a 1-0 home reverse to Hearts and also ended the match with nine men.
Hearts condemned The Dons to their fourth consecutive defeat - and fourth match without a goal - courtesy of a 12th minute own goal from Steve Tosh.
Paul Hartley's corner was headed on by Kevin McKenna and Tosh managed to divert the ball into his own net.
Tosh was later sent off for two bookable offences on 78 minutes while ten minutes previously, David Zdrilic was dismissed after collecting a brace of yellow cards.