Hampshire 187 and 178 for 8 (Gleeson 3-45) lead Lancashire 273 (Jones 68, Edwards 4-78) by 92 runs
Lancashire have been relegated to Specsavers County Championship Division Two after they failed to secure a third batting bonus point against Hampshire, although they remain on course for victory.
Lancashire needed to reach a minimum of 300 and beat Hampshire, while elsewhere requiring Nottinghamshire to fall short 200 - which they did - and then lose to Somerset at Trent Bridge. But after Fidel Edwards blasted through the tail with the second new ball they fell 27 runs short of the 300 mark and their demotion to the second tier was confirmed.
Next season will therefore be the first time since 2015 that Lancashire will play their cricket in Division Two. On that occasion they only spent one year in the second before they were immediately promoted back to Division One.
They join Worcestershire in dropping down this season, although they remain on top of their clash at the Ageas Bowl, restricting Hampshire to 178 for 8 - a lead of just 92.
Rob Jones had brilliantly batted Lancashire in the right direction to stay up with a morning half-century. Jones, who had come into the game with an average of less than 10 in seven Championship innings this season, added 84 runs with Dane Villas for the sixth wicket, after Liam Livingstone had been yorked by Edwards, his 50th scalp of the season, in the second over of the day.
But after a classy, attacking fifty, Jones departed two overs after lunch when Ian Holland had him lbw - the Australian's third wicket in figures of 3 for 48.
That left Lancashire needing 59 from the last three wickets, and Josh Bohannon and Tom Bailey kept the score ticking until the second new ball. But on 273, Kyle Abbott and Edwards reconvened and took three wickets between them to send the visitors down.
Abbott needed just three deliveries to find Bohannon's outside edge, with Joe Weatherley taking a good catch at third slip. Former West Indies fast bowler Edwards then sealed Lancashire's fate by bowling Bailey with a fierce full ball before Saqib Mahmood was lbw two balls later.
The difficulties of batting on a bowler-friendly pitch, despite blue skies above, were proven by Hampshire's upper order. With a first-innings deficit of 86, Hampshire lost Oli Soames to a fine head-high catch by Vilas at second slip, before Joe Weatherley was caught behind - both off the bowling off Richard Gleeson.
That brought Jimmy Adams to the crease for his final innings in first-class cricket, accompanied by warm applause from the crowd and away team players. He rode his luck when he was dropped by Jones at second slip on 2, before departing lbw for 13, and ending on 14,145 first-class runs.
Adams was given a guard of honour by his Hampshire team-mates before hugging his father on his way to the dressing room for the last time as a professional batsman.
Bohannon then took three wickets to leave the hosts 88 for 6. The bowler had Sam Northeast lbw to the fifth ball after tea, James Vince caught behind and then, a ball later, Holland bowled.
Liam Dawson and Lewis McManus added 43 for the seventh wicket to start a recovery, before the former was strangled down the leg side. Gareth Berg was caught at second slip but Kyle Abbott and McManus guided Hampshire to the close eight down.