Southampton improved on recent outings to demolish Tim Sherwood's Aston Villa side at St. Mary's ahead of a final day meeting with Manchester City.
Saints took the lead after just 13 minutes, when Mané raced on to a flick on from Graziano Pellè. The Senegalese forward's initial attempt was parried, with Mané stroking the ball over the line with his left-foot at the second attempt.
His second followed swiftly, as he rushed on to a loose ball after Given couldn't keep hold of a poked Shane Long attempt, lifting the ball over the line from a tight angle.
His third goal in a record two minutes and 56 seconds came when Long's direct run in the left channel saw him square the ball for Mané, who side-footed a first-time attempt high into Given's goal.
This saw Mané obliterate the record for the fastest Premier League hat-trick, previously held by Robbie Fowler after the Liverpool striker's treble against Arsenal as an 19-year-old in August 1994, which was registered in four minutes and 32 seconds.
Mané's hat-trick also set a record as the fastest top-level hat-trick in English football since the end of World War Two.
The fourth goal of the afternoon came in the 26th minute, when Long reacted to place the ball in the Villa net after Given failed to keep hold of an effort from Pellè.
Four became five before long, with Long netting again - although this time in more spectacular fashion.
The Irish forward robbed Ashley Westwood in midfield, picking up the ball and taking a touch before curling the ball over Villains keeper Given from all of 40 yards.
There was a sixth goal in the final minute of three additional minutes at the end of the first half, as Christian Benteke towered over Saints captain José Fonte to place a header to perfection, leaving Paulo Gazzaniga grasping at thin air.
Long dragged a volley just wide of goal at the start of the second half after the visitors failed to deal with a James Ward-Prowse corner, before firing wide again after a ball in from Pellè in the 61st minute.
Maya Yoshida replaced Steven Davis in the 72nd minute as Ronald Koeman made his first change.
Sherwood had already made two changes, withdrawing Charles N'Zogbia and Leandro Bacuna in order to send on Gabby Agbonlahor and Matthew Lowton by the 67th minute.
Yoshida almost announced his arrival with a goal, but for a last-gasp save from Given on his goal-line after the Japanese defender wriggled free at the far post on a Ward-Prowse corner. <!-- ######## START FLOATED VIDEO SNIPPET ######## -->
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A low ball in from Mané eventually rolled into the path of Nathaniel Clyne, only for the Saints defender to blaze over.
Koeman made his second change in the 79th minute, sending on Filip Djuričić on to replace Ward-Prowse.
Djuričić played a key role in the 6th goal for Saints, driving at the Villa defence before playing a one-two with Mané, using a back-heeled pass to set the Senegalese forward free.
He then curled a cross into the path of Pellè, who volleyed past the helpless Given to record his 16th goal of the season since joining from Feyenoord in the summer.
With four minutes of regulation time remaining, both bosses made their final change of the afternoon. Sherwood replaced Jack Grealish with Scott Sinclair, as Koeman gave Eljero Elia a run-out in place of Pellè.
Villa had a late chance when Agbonlahor forced a save out of Gazzaniga in the 93rd minute, but Saints held on to record a 6-1 win.
Saints remain 7th in the league table, but should they beat Manchester City next weekend, and both Liverpool and Spurs fail to win their own games at Stoke City and Everton respectively, Koeman's men could finish 5th in the Premier League.