Alasdair McKee's men entered the game needing a win having demolished Connacht 62-6 at Larne RFC before agonisingly losing 10-9 away to Munster last week.
The hosts started well and Ballyclare outside centre Gary Clotworthy was successful with two of three early penalty attempts to give his side a 6-0 lead.
A fine break by visiting second row Niall Smullen sent fullback Stephen MGee in under the posts for a fine try. Bill Duggan converted to put Leinster 6-7 up.
Clotworthy missed another three pointer but he was to get five shortly after when he rounded off an excellent counter attack. Cooke fullback Ashley Finlay (pictured) made a searing break down the left touchline before finding clubmate Phil Cartmill, who linked with Jason Morgan to put Clotworthy over.
The next period of play was highly eventful with Morgan joining three Leinster players in the sin bin. Clotworthy took advantage with a penalty on the stroke of half time to take Ulster into the break with a 14-7 lead.
Ulster had enjoyed a dominant scrum in the first half and that continued after the interval; no8 Jamie Johnston capitalised on one such set piece to crash over after 45 minutes. Morgan added the simple conversion to extend the lead to 21-7.
But Leinster showed great fight to work their way back in the game with two converted tries in the last quarter, levelling the game at 21-21.
Clotworthy had another penalty opportunity, which would have handed Ulster the win and the championship title but his effort drifted just wide.
Overall it was a promising campaign for Ulster, whose points difference of +55 over the three games highlights their ability.