Clerc, who scored a last-minute try in February for France to beat Ireland 20-17 at Croke Park and retain the Six Nations title, now has five tries in the tournament after once again proving to be Irish team's nemesis.
Ireland had only beaten France once in Paris since 1972, and that statistic never looked like changing after Elissalde had kicked four straight penalties for a 12-3 halftime lead.
Clerc's two second-half tries added polish to the victory.
Elissalde, who looks to have formed a dangerous new partnership with flyhalf Frederic Michalak, was busy around the fringes and gave good service behind his dominant forwards.
France led 3-0 in the seventh minute when Elissalde kicked a 20-metre penalty after catching Ireland's forwards offside from a quick tap.
Flyhalf Ronan O'Gara missed a chance to level the score in the 11th from a 40-metre penalty, and Ireland's luck got worse just three minutes later.
Having pinched possession from Michalak just outside his 22, winger Andrew Trimble took an inside ball to slice through a hole and into space, but he managed only to run into referee Chris White, who needed treatment for a bloody nose.
France almost scored in the 16th minute after Heymans made a break down the left flank.
The ball was then spread to the far wing for Poitrenaud to dive over in the corner, but the TV official ruled Irish winger Shane Horgan had taken his opposite number into touch at the crucial time.
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However, France continued to press ahead as Elissalde landed two penalties to extend their lead to 9-0 by the 22nd minute.
The usually reliable Irish lineout couldn't keep its own ball, while its kicking game and the forwards' ability to slow down their opponents' possession at the breakdown were second best all night.
O'Gara finally got his side on the board when he steered a 35-meter drop goal in the 37th, but Elissalde then kicked another penalty for 12-3 at halftime.
Second half surge
France's forwards surged after the break, but in one drive center Damien Traille couldn't hold a pass as he ran into a gap in the 22 after more sharp lead-up work from Elissalde.
Having replaced lock Sebastien Chabal with Lionel Nallet, France moved out to 15-3 as Elissalde kicked a penalty in the 56th minute after France went into Ireland's 22 following Simon Easterby's head-high foul on Michalak.
With the Irish scrum having just been splintered to force another scrimmage, Michalak cleverly kicked with the outside of his right foot for Clerc to score unopposed as the ball held up in the in-goal in the 59th minute.
Elissalde missed the conversion to leave the score at 20-3.
But Ireland's problems got worse when lock Paul O'Connell was sent to the sin bin for an infringement at a lineout and that effectively ruled out any chance of a late comeback.
With 12 minutes left, Clerc scored his second try when he caught a short chip from Elissalde and then wrestled his way through two attempted tackles to go over in the right corner.
Elissalde missed the conversion, but the job was done.