Having hit the woodwork three times in the first-half and been frustrated by the brilliance of UCD keeper Conor O’Donnell, Rovers finally found a way to goal on 57 minutes.
Former student Finn cut in down the inside-right channel and drilled a powerful shot that O’Donnell got a hand to but couldn’t keep out.
His goal under siege, O’Donnell made a parry save from a Brennan header and then somehow pushed Gary McCabe’s drive out for a corner.
But the 20-year-old keeper could do little to stop Rovers’ second stunning strike on 85 minutes.
Stephen McPhail, who started a league game for the first time, back-heeled the ball to McCabe to set up Brennan who found the top corner of the net from 20 yards.
Save for an early chance for Dean Clarke, Rovers owned the first-half, striking the woodwork three times and forcing two sublime saves from O’Donnell.
O’Donnell had the frame of his goal to thank for the first time on 10 minutes as Finn’s cracking drive from distance rattled his crossbar.
With Rovers well on top, the woodwork came to UCD’s rescue again on 28 minutes as the busy McCabe set up Sean O’Connor to drill a shot off the butt of O’Connor’s right-hand post.
A minute later, Rovers should have gone ahead. Ciarán Kilduff nodded Simon Madden’s cross down perfectly into the path of Finn who skied over the top form ten yards.
A combination of O’Donnell and his other post somehow kept Rovers out on 34 minutes.
O’Connor set up McCabe inside the area and his shot was arrowing to the top corner before O’Donnell pushed it onto a post. McCabe smashed the follow-up into the side netting.
And O’Donnell was there again to make a terrific one-handed save to keep out Jason McGuinness’s downward header from a McCabe corner as UCD somehow survived until the break.
SHAMROCK ROVERS: Murphy; Madden, Kenna, McGuinness, Byrne; Brennan, McPhail; McCabe (Sheppard, 89), Finn (Bayly, 89), O’Connor (Waters, 78); Kilduff.
UCD: O’Donnell; Barker, Douglas, Boyle, Matthews; Mulhall (O‘Halloran, 75), Creevy, Benson, Clarke (Belhout, 78); Burke (Kavanagh, 67); Morrison.
Referee: Graham Kelly (Cork).