Templeton held off the challenge of three would-be tacklers as he cut in off his left wing before tucking in the 19th-minute opener for the visitors and Stephen Elliott netted a second from close range after 67 minutes.
Captain Derek Riordan was sent off after 88 minutes for a crude challenge on Rudi Skacel as Colin Calderwood's first home match in charge ended in defeat, leaving Hibs alongside St Mirren at the foot of the SPL table.
Referee Willie Collum was in charge of a second high-profile derby in a matter of weeks, but avoided controversy after being criticised for his officiating in last month's Old Firm derby by Celtic manager Neil Lennon.
However, there were two flares thrown on to the pitch and numerous coins in incidents which may be brought to the Scottish Football Association's attention.
Lennon was in attendance at Easter Road and will be fully aware of the danger posed by Templeton when his side travel to Tynecastle on Wednesday.
After Hibs - unchanged from the 1-0 loss at Dundee United - began on the attack without threatening Marian Kello in the visitors' goal, Templeton issued a warning for what was to come.
The 21-year-old winger - one of three changes to the starting XI from last week's 3-0 loss to Kilmarnock - cut in off the left flank and fired a right-foot shot from 25 yards straight at Mark Brown in the Hibs goal as the visitors had their first opportunity after 16 minutes.
But three minutes later he had put Hearts in front with a stunning solo goal.
Templeton picked up the ball in the left channel and shrugged off the challenge of Danny Galbraith, Michael Hart and Sol Bamba as he cut in on a diagonal run.
His pace carried him into the area before he tucked a right-foot finish low into Brown's goal.
The goal was greeted with Hearts fans throwing two flares on to the pitch, with proceedings delayed for around two minutes before they were extinguished by stewards.
Hearts looked the more likely to add to their advantage, but as Templeton's pace troubled Hibs once more, his run was brought to an abrupt end by De Graaf, the Hibs midfielder booked for the cynical foul.
Elliott then raced on to Kevin Kyle's through ball, but could not get the decisive touch under pressure from Jonathan Grounds and Paul Hanlon.
John Rankin spurned numerous free-kick opportunities, while Hibs striker Riordan opted to take the final set piece of the first half and too failed to find a team-mate.
Rankin was replaced at the interval by Lithuania striker Valdas Trakys, but still Hibs' set-pieces were poor, with Kello untroubled.
Kyle and Trakys wrestled with each other in the area following a Riordan free-kick with Hearts winning possession and Collum adjudging the challenge to be fair.
Skacel then shot acrobatically off target from Black's centre, with Elliott also looking to play the ball as Hearts spurned the chance of a second after 57 minutes.
David Wotherspoon replaced Galbraith as Calderwood looked for fresh attacking impetus after 66 minutes.
But immediately it was Hearts who scored as Kyle knocked down a long ball forward, Templeton centred and Elliott netted into the bottom corner.
Ismael Bouzid headed wide from Eggert Jonsson's cross as Hearts continued to press forward.
Kyle was booked for throwing the ball into an advertising board after winning a throw-in and Collum was handed what appeared to be a coin thrown from the Hibs section of fans.
Ryan Stevenson was introduced for Templeton after 80 minutes, with Hibs fans also beginning to leave the action, their side seemingly short of ideas.
Ian Murray was introduced for Lewis Stevenson as Calderwood made his third change after 84 minutes.
Elliot came on for goalscorer Elliott with four minutes remaining before Hart was penalised for hand ball after grappling with Ryan Stevenson on the left wing.
Riordan was shown a straight red card for a terrible challenge which pole-axed Skacel two minutes from time.
More coins were thrown as Skacel received treatment as Hibs were left to contemplate a record of two wins from 14 games this season.