It was the Wales international's first game back following a three-match ban and he rewarded Gary Megson's faith in him by grabbing the crucial goals to ensure the League One side avoided an embarrassing FA Cup first round exit.
Backed by a riotous home support of more than 6000 fans, the largest gate the Wessex Stadium has ever held, the Conference minnows so nearly took the lead on the stroke of half-time when top scorer Chukki Eribenne struck a post.
The former Bournemouth striker broke free from Forest captain Ian Breckin and delivered a stinging left-foot drive that beat Paul Gerrard but not his left hand post.
A full 66 league places separated the two sides before kick-off but after the opening exchanges it was hard to notice the difference.
However, Forest slowly began to stamp their authority on the second half of the game and were denied on 54 minutes when a John Curtis free-kick was punched away by Jason Matthews.
Curtis then saw a long-range effort drift wide.
The visitors took the lead after 67 minutes when Nicky Southall broke down the right and delivered a low cross for Taylor to fire home.
Just minutes later Weymouth nearly drew level. Lee Elam, the hosts' chief tormentor, stole a yard on Breckin and crossed, only for his centre to turn into a shot that had to be parried by Gerrard.
This proved to be the home side's last real sniff at goal as Forest doubled their lead in the 73rd minute.
In similar fashion to the first, James Perch crossed from the right and Taylor was again on hand to tap home and send Forest into the second round.