The Yorkshire One side fell behind early and left themselves too much to do in the second half.
Brighton took the lead early on with winger Steve Dean finishing well after a neat final pass, Finch adding a difficult conversion. Mistakes prevented Bridlington from gaining any momentum and they lost centre Ryan Girking to the sin-bin on 25 minutes.
With the man advantage, Finch added two more penalties before finding a gap in the defence, running half the length of the pitch and breaking tackles to touch down under the post before adding the extras.
After reducing the mistakes, Bridlington hit back with a drop goal from Jamie Muro followed by a try from full-back Sizwe Zondo, who found space on the wing from an earlier charge down.
Zondo added his second after fending off and dummying through the Brighton defence, Stevie Mellonby converting from the touchline to give Bridlington a sniff at 20-15.
But Brighton captain Will Smith smashed his way through to go over the line and, with the conversion, his side now led 27-15.
Bridlington, however, cut the lead again with a well-worked Tom Brown converted try before Finch edged Brighton home with a penalty.
Bridlington captain Alistair Sutton said: “If we had played like we did in the second half and played to our game plan in the first 20 minutes I feel we could have walked away as champions. But you can’t give a team a 20-0 lead.”
Bridlington: Musawenkosi Zondo; Mellonby, Kemp, Girking, Burrows; Muru, Robinson; Luxton, Kinnaird, Dixon, Murray, Cappleman, Heeley, Sutton, Thompson. Replacements: Waines, Warley, Kerr, Edmond, Hanson, Brown, Davies.