Worsfold, who had earlier hit the post, opened the scoring six minutes before half-time.
The points were made safe in injury-time when Alex Simmonds made it 2-0.
There was still time for the home side to grab a goal, but Trinity hung on for an away win — a rare commodity last season.
Boss Lee Sinnott said: “It was good to tick that box in a new season. You want productive away form to supplement home form.
“It was a battle so I am delighted for the players.
“ And it was lovely to see the supporters there and Gainsborough shirts around the ground.”