Dawson, who played the last of his three England Tests in 2017, grabbed his first wicket of the second innings when Matty Hurst missed an attempted big hit and was bowled for a 19-ball duck to leave the score at 44-4.
Wells battled hard to reach his half-century off 143 balls but when he was unluckily bowled by Dawson, playing on to one that spun sharply out of the rough, the result was inevitable with Lancashire 88-6.
The 34-year-old then claimed three of the final four wickets, including his 10th of the match when Tom Bailey was caught off a skier to wrap up the win with a day to spare.
Hampshire jump up to second in the table, albeit with other matches yet to finish.
While Dawson rightly deserves the headlines, it was another disappointing defeat for Lancashire on the back of losing to leaders Surrey by an innings and 63 runs at The Oval.
They are eighth in the table, but will fall to ninth when Nottinghamshire complete their match, with only three games left to save their Division One status.
It looks increasingly likely that the match away to Worcestershire in the final round of games starting on 26 September could prove to be decisive in that battle.
