Jacks claimed figures of 2-29 and scored 65 off 39 balls but when he was dismissed to leave Surrey at 116/5, Sussex looked to be in the driving seat before 40 from 29 balls from Overton snatched victory for Surrey.
The hosts won the toss and stuck Sussex into bat with the Sharks looking to put themselves another win closer to qualification for the quarter-finals and were helped by the return of Phil Salt from England duty.
However, Salt was unable to do any major damage as he was trapped leg-before for one in the second over by Jacks before the visitors then lost Aaron Thomason two overs later, caught by Jacks off the bowling of former Sussex man Reece Topley for five.
After a nightmare start, Luke Wright and Calum MacLeod settled things down for Sussex as they ended the powerplay on 44.
However, just as the Sharks looked to accelerate, MacLeod was dismissed for 22, bowled by a superb yorker from Liam Plunkett to the leave the visitors teetering at 57/3.
Delray Rawlins joined Wright but he could only add 33 runs with the Sussex captain before Surrey broke through again to dismiss the Bermudian, with Jacks having him caught by Plunkett for 20 after having being hit for a huge six by the 23-year-old a ball previous.
Sussex were soon in real trouble when Wright departed for 45 after being bowled by Rikki Clarke to leave his side 108/5 with just five overs to go.
It was to be David Wiese who got the Sharks out of trouble once again as he and Ravi Bopara added a quickfire 43-run partnership in just over three overs
Bopara had been out of form since his arrival at Hove, scoring just 66 in seven innings but this didn’t seem to affect him when he struck three consecutive sixes off Gus Atkinson in the penultimate over.
Atkinson had his revenge the following ball when Bopara edged through to Ben Foakes but the all-rounder had certainly played his part as he finished on 24 off just nine balls.
Wiese had been his usual imperious self but fell in the final over for 33 off 23 balls, caught via a relay catch by Atkinson off Topley before George Garton was able to add some useful runs as Sussex finished on 165/7.
The Sharks prevented Surrey from getting off to a fast reply in the first four overs of the powerplay and were duly rewarded for their good bowling with a wicket, Hashim Amla caught by MacLeod off Ollie Robinson for 13.
The game was evenly poised at the end of the powerplay as Surrey also finished on 44/1 but with Jacks looking set after taking a large majority of the strike.
This strike-farming deprived his partners of any action and as Sussex continued to bowl tightly this pressure eventually tolled with Foakes only able to add five runs in a 31-run partnership with Jacks before he was bowled by Will Beer.
Rory Burns was able to contribute 10 runs before he was dismissed in the unluckiest of ways when Rawlins was able to fingertip a Jacks’ drive back on to the stumps to see the England Test opener caught short at the non-striker’s end.
Jacks’ partners may not have been able to contribute much, but the Surrey opener was certainly able to himself as he raced to a half-century off 32 balls which included seven boundaries.
He continued to power on, adding two more sixes to his boundary count before he was bowled by Beer for 65 as he attempted a big sweep against the leg-spinner.
Surrey were soon under the pump and their task of chasing 168 became harder just 16 balls later when Beer struck with the final ball of his allocation as Clarke was caught in the deep by Bopara for seven to leave the hosts on 116/5 at the end of the 14th over.
Sussex looked to be favourites as the game headed into the final overs, but Smith and Overton got Surrey back in contention as they added 34 runs in four overs to leave the hosts needing 16 runs from the final two overs.
Overton then all but sealed the game as he smashed 13 runs off five balls in the penultimate over bowled by Robinson to leave Surrey in touching distance of victory but the Sussex star was able to take it to the final over when he dismissed Smith for 16 after the batsman slapped a full-toss straight to Wright.
There was to be no final over drama as Overton finished the game by clattering a four off the first ball bowled by Wiese to end a superb knock from the former Somerset all-rounder as Surrey won by four wickets.
The defeat for Sussex now leaves them two points adrift of the automatic qualification spots and three points away from the top spot.
The Sharks will have to reassess ahead of their now crucial game versus Middlesex at Hove on Friday with the Lord’s-based outfit having picked up a two-run win earlier in the day which has now taken them within a point of Sussex.
Another Sussex loss will see them go into the final game of the season versus Essex needing a win and them banking on Middlesex to lose, but a victory over the London county will see them clinch third spot.