They led Ulster 9-3 at half-time with Seapoint's ever reliable out-half Brian Keegan kicking three penalties to a single effort from Ulster number 10 John Semple.
Keegan, his province's vice captain, was particularly influential in the wins over Munster and Connacht, scoring 24 points.
Ulster, who lost 6-3 to Munster last time out, edged closer to Leinster with a second penalty from Limavady clubman Semple.
But the game's defining moment came soon after when Karl Manning intercepted a pass in his own 22 and the Navan full-back turned on the after-burners to scorch over for a converted try.
Ulster piled on the pressure in the final quarter and their flanker and captain Peter Kelly succeeded in crossing the visitors' whitewash, with Semple converting.
But that late try was not enough as Leinster hung on to see out the championship unbeaten, ending their five-year wait for the Junior Interprovincial title.
Afterwards Boyne clubman Eoin Collins was presented with the championship trophy by Ulster Branch President David Workman.
In Saturday's other game, Munster scored a comprehensive 29-0 win over Connacht at Richmond RFC in Limerick.
Ger Looby, Stephen O'Brien, team captain Gordon Ryan and David Joyce scored a try apiece with Cobh Pirates out-half Eamon Ronan kicking three conversions and a penalty.
Despite failing to defend their title, this was an encouraging championship for Munster with a number of players getting capped at this level.
Those new caps included two of Saturday's try scorers - Clonmel winger Joyce and openside flanker Looby from Kilfeacle.