Rangers manager Ally McCoist admits he and his team are fighting for their futures after the defeat to Annan.
Goals early in the second half from Ally Love and David Hopkirk sealed a famous victory for the visitors, the first under manager Jim Chapman.
A close range effort from Andy Little got Rangers back into the game but the home side couldn't find another goal.
McCoist described the performance as "unacceptable" while Chapman hopes the result can give Annan renewed belief.
A largely forgettable first half was punctuated by some good goalkeeping.
Alex Marshall kept out a wonderful long-range effort from Rangers' right-back Anestis Argyriou, before left-back Lee Wallace also tested the visiting keeper's reflexes with a shot from a corner.
At the other end, Neil Alexander did superbly well to keep out a deflected shot from David Murray.
Before that, Rangers striker Dean Shiels was stretchered off with a knee injury and replaced by Barrie McKay. A scan on Tuesday will determine the severity of Shiels' problem.
The game sparked into life after the restart with three goals in twelve minutes. Love stunned a big Ibrox crowd with a neat backheel to open the scoring for Annan from Daniel Orsi's cross two minutes after half-time.
Eight minutes later, Athletic doubled their lead when Hopkirk met David Murray's cross and home keeper Alexander failed to keep the ball out.
Little replied almost instantly for Rangers when he reacted quickly to substitute Fran Sandaza's knockdown from a corner.
Plenty of Rangers pressure followed, with Lee McCulloch causing Annan lots of problems and drawing two fine saves from Marshall.
Annan had a chance to extend their lead but Alexander turned away a good long-range effort from Hopkirk.
Earlier in the week, Rangers chief executive Charles Green described the current side as "probably the worst" Rangers team ever.
The players did nothing to contradict that withering assessment against Annan, whose small band of fans celebrated a result, and performance, they'll never forget.