Set 240 to win, Michael Klinger (53) and George Hankins (67) added 99 for Gloucestershire's second wicket.
Taylor then smashed 64 from just 36 balls before being out with the scores level, but the visitors sealed victory.
Luke Wright had earlier made 84 for Sussex, but was caught behind off the impressive Chris Liddle (5-52).
After Wright fell on 182-5, having earlier been aided by Laurie Evans' 48, Sussex's batsmen struggled, with only Jofra Archer (30) making any real impact on the scoreboard.
Both sides are now on three wins after seven matches, but Sussex have a point extra after their rain-ruined match against Middlesex.
They can still make the top three - and a place in one of the two quarter-finals if they beat Hampshire at Southampton on Wednesday and Surrey lose to Gloucestershire at Bristol.
Gloucestershire's Jack Taylor:
"I'm hitting the ball well at the moment and it was nice to get us over the line for a good victory. We've just been too inconsistent in this format but we played well today.
"We weren't sure what a good score was going to be but we bowled well, especially Chris Liddle, and then Michael Klinger and George Haskins set things up nicely for me and Ian Cockbain at the end.
"Although we can't reach the knockout stages it's important to keep playing good cricket, not least because a lot of the skills are used in T20. It's good practice for that."
Sussex head coach Mark Davis:
"It's incredibly frustrating because we have played a lot better in this format but we have twice got ourselves in good positions and not won either game.
"We thought we had a competitive score but we got ourselves into a good position and then kept losing wickets at crucial times. They then set the game up with the way Klinger and Haskins batted.
"If we win the last game we'll give ourselves a chance of making the knockout stages so there is still a lot to play for."
Jack Taylor's fourth One-Day Cup half-century of the season helped Gloucestershire to a six-wicket win over Sussex with eight balls to spare.