Andy Little was also on the score-sheet in a match which saw Lee Wallace sent off in the second half and the visitors fail to level from the penalty spot.
The win allowed Rangers to move six points clear at the summit of the Irn-Bru Third Division.
Francisco Sandaza made his long-awaited comeback from injury as Rangers made two changes.
The Spaniard had been out of action since suffering a broken cheekbone against Motherwell at the end of September but he started at Ibrox tonight, along with Barrie McKay.
Kevin Kyle and Dean Shiels both missed out after being rated doubtful following the weekend win over Montrose.
Rangers chief executive Charles Green took to the pitch to address the crowd ahead of the match and confirm that a total of £22.2million had been raised through a share issue.
However, while his speech was taking place, a section of the support staged a protest against potential plans to sell the stadium naming rights.
The match kicked off almost on time after the Annan team bus broke down en route to Glasgow and the visitors arrived at the stadium later than planned.
Rangers looked dangerous from the outset and McKay saw his shot blocked by goalkeeper Alex Mitchell, before Sandaza fired wide from the rebound.
McKay then supplied a cut-back for Little but the Northern Irishman's strike cannoned off the body of Michael McGowan.
Mitchell was called into action again when he produced an impressive save to tip a thunderous 20-yard effort from Templeton just over the bar.
There was proving to be little respite for Annan and Lee McCulloch was just off-target after meeting a corner, before a shot from Little came off the goalkeeper's legs.
The pressure paid off after 29 minutes when Rangers surged into the lead thanks to a lovely low drive from Templeton from just outside the area that caught the opposition defence unaware.
The winger could have claimed his second of the night when he threw himself in front of a Wallace cross only to see his header scooped to safety by Mitchell.
Annan's best chance so far came just before the break when Chris Jardine blasted narrowly wide, before Lewis Macleod's header forced another decent save from Mitchell at the other end.
Annan were handed the chance to haul themselves back into the match amid dramatic scenes in the second half.
Wallace was shown a straight red card by referee Paul Robertson, who ruled the full-back hauled down Jack Steele in the box, while skipper McCulloch earned himself a booking for dissent.
Scott Chaplain stepped up for the spot-kick but goalkeeper Neil Alexander did well to smother with 60 minutes on the clock as Rangers retained their lead.
The dismissal resulted in striker Sandaza being replaced by defender Ross Perry, who made his return following a hernia operation.
The home side then doubled their advantage after 64 minutes when Little raced onto a Templeton pass before sending an angled drive past Mitchell and into the far corner.
Templeton added to his own tally six minutes later with a low shot from 12 yards, shattering any hopes of an Annan comeback.