Ally McCoist's team dropped points for a third successive away game and suffered another blow when David Templeton was carried off with an ankle injury.
Rangers again struggled to impose themselves on part-time opponents and home midfielder Jack Steele missed a great chance early in the second half.
Rangers only exerted superiority in the final quarter and even then lacked penetration and Alex Mitchell secured the point for the hosts with a stunning last-minute save from Lee McCulloch.
Annan were well-marshalled by imposing centre-backs Steven Swinglehurst and Peter Watson on their artificial pitch and their hard-working and skilful midfield matched Rangers for much of the game.
Galabank housed a record attendance of 2,517, enabled by the summer installation of terracing behind a second goal.
Chairman Henry McClelland rubbished the predictions of Scottish football's impending doom that accompanied votes on the destiny of the newco Rangers by writing "we are all loving Armageddon" in his programme notes.
He also reflected that the Rangers fans he had spoken to in the build-up to the game were falling in love with football again but they might need some away victories to retain such affection following previous draws at Peterhead and Berwick.
Their latest journey into the unknown took them to a town with a population of 8,300. Annan were the newest entrants to the Scottish Football League before Rangers, winning admission in 2008 after neighbours Gretna went into liquidation.
The club, who previously played in the East of Scotland League, accommodated French and German television crews with much of the press seated on chairs from the social club just next to the corner flag as they squeezed as many fans as possible into their one stand.
Rangers, who were without Andrew Little and Anestis Argyriou through injury, suffered another blow after five minutes when Templeton crumpled holding his left ankle without anyone near him as he ran with the ball down the wing.
Francisco Sandaza eventually came on as Templeton was carried away on a stretcher.
There was little meaningful goalmouth action in an even opening 20 minutes and Annan had the first real chance when Michael McGowan's cross found its way to Michael Daly in space inside the box but the centre-forward could not get the ball down quickly enough and the visiting defence managed to scramble the ball to safety.
Daly and McCulloch headed over after corners at either end and Annan survived a counter-attack when Steele produced a last-ditch sliding tackle to stop Barrie McKay converting fellow teenager Lewis McLeod's ball over the top.
Mitchell made his first save in the 29th minute when he got down to save from Sandaza after Dean Shiels had found the Spaniard's run.
McCulloch volleyed wide from Sandaza's cross but Annan were still getting men forward and a corner from Graeme Ramage flashed across the Rangers goalmouth.
Rangers were almost exposed in the final minute of the half when Steele fired a bouncing ball just wide and McCoist looked an angry man as he marched towards the tunnel despite Shiels seeing a shot beaten away by Mitchell in injury-time.
Alexander made the first save of the second half, from Daly's header, and the hosts had a great chance in the 55th minute.
Ramage was brought down as he went to collect Scott Chaplain's pass but Steele ran on to the loose ball to leave him through on goal but the midfielder dragged the ball wide from 15 yards.
Alexander exchanged words with fans behind his goal, apparently frustrated by criticism, and the away support's upset was audible when Sebastien Faure hit two crosses into their ranks in quick succession.
Rangers did begin to create chances after the hour mark. McCulloch had a scuffed effort cleared in the goalmouth and Shiels chipped the ball harmlessly into Mitchell's arms after a good pass from McKay.
Mitchell blocked Sandaza's shot with his feet after the Spaniard cut in from the left. Rangers found an urgency, or perhaps desperation, to their play and pushed their opponents back for much of the final quarter.
Both Sandaza and Lee Wallace were both off target from long range but they struggled to carve out a clear chance until the ball fell for McCulloch 15 yards out following a long free-kick.
The striker caught it on the half volley and the ball seemed destined for the top corner but Mitchell somehow threw himself across goal to touch it over the bar.