The Argentine, deputising for Girvan Dempsey at full back and taking over kicking duties from fly-half David Holwell in wet and windy conditions, succeeded with three of five attempts at the Ulster posts.
Leinster have now not lost at home in eleven league outings - stretching back to last March.
Mark McCall's visitors had earlier run up an 8-0 half-time lead, thanks to number 10 Adam Larkin's third minute penalty, and an opportunist try from Ireland wing Tyrone Howe in first half injury time.
Howe intercepted the returning Gordon D'Arcy's midfield pass before hacking on three times and evading the late challenge of substitute Brian O'Driscoll - one of six Ireland internationals on Declan Kidney's bench - to buoy Ulster's hopes of a first win in Dublin since 1999.
Poor discipline cost them in the second half however, as with both O'Driscoll and Howe sin-binned for an off-the-ball altercation, Contepomi kicked Leinster back to 8-6 on 43 and 54 minutes.
The 27-year-old South American was offered two late chances to win the game as Ulster replacement prop Ronan McCormack was caught offside with three minutes remaining, and another of McCall's substitutes Rowan Frost was yellow carded at the death for killing the ball.
Contepomi took the second, straighter opportunity to see Leinster move seven points behind league leaders the Ospreys, back in third.
Leinster meet Lyn Jones' pace-setting side at Donnybrook, in early March.
Monday, December 27Leinster 9 Ulster 8, DonnybrookScorers: Leinster: Pens: Felipe Contepomi 3Ulster: Try: Tyrone Howe; Pen: Adam Larkin
(15) Felipe Contepomi(14) Shane Horgan(13) Gordon D'Arcy(12) David Quinlan(11) Kieran Lewis(10) David Holwell(9) Brian O'Meara(1) Reggie Corrigan (Capt)(2) David Blaney(3) Ricky Nebbett
Replacements used: Brian O'Driscoll for Quinlan (32 mins), Shane Byrne, Malcolm O'Kelly and Eric Miller for Blaney, Gissing and McCullen (all 63), Denis Hickie for Lewis (71). Not used: Emmett Byrne, Brian O'Riordan.
(15) Bryn Cunningham(14) Tommy Bowe(13) Paul Steinmetz(12) Kevin Maggs(11) Tyrone Howe(10) Adam Larkin(9) Kieran Campbell(1) Simon Best(2) Rory Best(3) Rod Moore(4) Gary Longwell(5) Matt McCullough(6) Campbell Feather (Capt)(7) Neil Best(8) Roger Wilson
Replacements used: Ronan McCormack for S Best (half-time), Rowan Frost for N Best (54), Nigel Brady for R Best (66). Not used: Matt Mustchin, Neil Doak, Paddy Wallace, Jonny Bell.
Referee: Alain Rolland (IRFU)