The Fife side have now gone seven league games without a goal - a new Premierleague record.
Dunfermline will argue they were denied a strong penalty claim late in the first half but Rangers did enough to merit the points.
David Weir made his debut at the heart of the Gers defence, which looked steadier than on their recent visit in the Scottish Cup.
Nacho Novo had a chance after barely a minute but he directed his shot straight at Dorus De Vries.
Dunfermline hit back and Stevie Crawford fashioned an opening for himself but Allan McGregor managed to beat his shot out.
Rangers moved in front in the ninth minute when Novo raced down the right and crossed to the back post for Adam to score from a tight angle.
De Vries then saved well from Barry Ferguson as Rangers continued to threaten with Novo looking lively.
Novo had another chance after half-an-hour but again De Vries was alert to the situation and made another good save.
Dunfermline were by no means out of it and finished the first half strongly.
The home fans felt they were denied a penalty when a Crawford shot appeared to strike the arm of Karl Svensson on the line, but referee John Underhill was not interested.
Ferguson had a chance to extend Rangers' lead early in the second half but sent a tame shot straight at De Vries.
Kris Boyd was denied by the Dunfermline goalkeeper in the 74th minute and Adam sent the rebound wide.
Then Ian Murray looked set for his first goal for the Ibrox club but his right-foot shot lacked pace and De Vries saved easily.
Jim Hamilton should have equalised in the closing minutes, but headed over from four yards after great work by substitute Adam Hammill.