Jones and Best both scored twice to lead Saints to their biggest home win of the season on a dramatic final day in front of a 32,000 sell-out at St Mary's.
Southampton needed to at least equal seventh place Stoke's result at QPR to be guaranteed a ticket to the end-of-season lottery.
Burley's side made a nervous start and found themselves behind after only 13 minutes when Southend captain Adam Barrett put the relegated visitors in front with his first goal since August.
Barrett silenced the home fans when he crashed in a fierce left-foot volley from the edge of the penalty area after some clever close control from winger Richie Foran.
Foran should have doubled Southend's lead just three minutes later when he put a free header over the crossbar from Simon Francis' cross.
Saints took until the 29th minute to really get going when Jones levelled with a back-post header after Gareth Bale had whipped in a teasing cross from the left touchline.
Trinidad & Tobago international Jones was denied a second when Southend's former Southampton goalkeeper Darryl Flahavan tipped his 20-yard piledriver on to the bar.
Jones continued to terrorise the visiting defence in the second half and his brave header in front of Flahavan allowed Best to put Southampton ahead with a close range tap-in with 48 minutes on the clock.
Best put the game beyond doubt ten minutes from time when he doubled his tally with a side-footed shot from another Bale cross.
The front pair combined again to add a fourth less than a minute later and erase any nagging fears that Saints would be overhauled by their play-off rivals.
Best raced on to a Danny Guthrie pass and was bundled over by Flahavan, but the impressive Jones was on hand to fire into an empty net just as referee Nigel Miller was about to award a penalty.