Lee McCulloch celebrated being handed the captain's armband with a brace of his own, as he netted his 100th and 101st career goals.
Dean Shiels was also on target, sparking the Rangers comeback with the equaliser after Jamie Duff had given Elgin a shock lead.
Templeton made his Rangers bow from the start as boss Ally McCoist made two changes from the Scottish Communities League Cup win over Falkirk midweek.
The winger was drafted straight into the starting line-up for the clash at Ibrox, ahead of Sebastien Faure, after sealing his move from Hearts late on Friday night.
Ross Perry was also given the nod as he took the place of skipper Carlos Bocanegra in the heart of defence, after the American joined Racing Santander on loan on transfer deadline day.
Minutes before the match was due to start, McCoist confirmed on the club's official website that McCulloch would replace Bocanegra as captain, describing the decision as a "no brainer".
Rangers enjoyed the best of the opening stages and came close after a couple of minutes when McCulloch fed Shiels and he dragged his shot narrowly wide.
Andy Little then raced onto a Templeton pass only to see his well-struck shot fizz just past the post as the home side continued the early pressure.
However, Rangers were hit with a sucker-punch when Elgin took the lead after 15 minutes when a corner broke for Duff and he lashed high into the top corner.
McCulloch could have levelled the scoreline again almost immediately but directed his header straight into the grateful hands of goalkeeper John Gibson.
But Rangers were back on level terms with 24 minutes gone when McCulloch's initial effort was blocked and Shiels pounced on the rebound to power home.
The Ibrox side then surged into the lead five minutes later when Ian Black teed up the shot with some fine play for Templeton and the new recruit rifled high into the net.
Templeton could have been celebrating again moments later when he unleashed from distance and Gibson was only just able to block for a corner.
The winger then tested the goalkeeper again with another decent effort but, this time, McCulloch was able to stab home the rebound from inside the six yard box to claim his century on the stroke of half-time.
Templeton claimed his second of the day just three minutes after the restart.
A Shiels corner into the packed box was collected by the winger, who turned and drove past Gibson from close range to leave Elgin with no way back.
There was also a double for McCulloch after 59 minutes. Little was the provider when he squared across the face of goal and a simple finish was all that was required from the new skipper.
McCulloch could have claimed his hat-trick when he met an Anestis Argyriou ball across the face of goal but blasted over, before firing wide after being set up by Barrie McKay.
There was no hat-trick for Templeton either, who was withdrawn to a loud ovation from the 46,015 crowd - the biggest in Scotland this weekend - to make way for Kevin Kyle late on.
He was later announced as the sponsors' man of the match following an impressive first outing for his new club.