Andy Little was also on the scoresheet 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 Third Division.
Rangers looked dangerous from the outset and Barrie McKay saw his shot blocked by goalkeeper Alex Mitchell, before Francisco 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.
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.