Andy Little pulled a goal back but it was a dismal day for the Third Division leaders.
After Charles Green's midweek statement that the current team is the worst in the club's history, the home players had a perfect opportunity to respond but in a first half, facing a chilling wind, the home side failed to convince or warm their fans. And in the second half it only got worse.
Rangers were keen to prove they were a better side that chief executive Green might think, but it took them 20 minutes to test the Annan goalkeeper. Anestis Argyriou's 20-yard drive had plenty of power but it was at the right height for Alex Mitchell to push the ball over the bar.
Rangers were forced into an early substitution after 32 minutes. Dean Shiels, in for the injured David Templeton, had his shot blocked by Chris Jardine, with Shiels' right knee taking the brunt of the challenge. Barrie McKay took his place wide on the left but before he could do any damage the Rangers goal had a lucky escape.
Hopkirk was the provider but his perfect cutback for David Murray was wasted as he tamely scooped the ball into Neil Alexander's hands from eight yards.
Rangers should have headed for the dressing room a goal down. Two minutes from the break Murray's shot took a wicked deflection off Emilson Cribari's outstretched leg and looked to be looping over Alexander until the goalkeeper recovered to squeeze the ball wide of his goal.
The home side were jeered off, but whatever manager Ally McCoist said in the dressing room it seemed to have no effect as two minutes into the second half Annan made the breakthrough. Danny Orsi swept the ball in from the left and Love nudged the ball past Alexander to silence the home crowd.
Francisco Sandaza replaced Kal Naismith as Rangers chased the equaliser but the Galabank side incredibly doubled their lead after 54 minutes. Murray sent a deep cross to the far post and Hopkirk bravely dived to head the ball low beyond Alexander.
Four minutes later McCoist's men reduced their arrears. McKay floated in the corner and although Sandaza's shot was heading towards a packed goal, Little headed it over the line for his 22nd goal of the season.