Craig Overton and Josh Davey sparked a dramatic Somerset resurgence at the Cooper Associates County Ground.
When the home side could add only 90 to their overnight score of 178 for 4, Steve Finn finishing with 5 for 77, they conceded a first innings lead of 89 runs.
But Overton (5 for 34) and Davey (3 for 28) skittled Middlesex for just 117 in their second innings, leaving a victory target of 207. By the close, the hosts had posted 104 for 4, with skipper Tom Abell unbeaten on 43.
Gloucestershire v Leicestershire
Bristol (third day of four): Leicestershire 421 & 125-3, Gloucestershire 275 - Leicestershire lead by 271 runs with seven second-innings wickets remaining
Leaders Lancashire are favourites to win their third LV=Insurance County Championship match of the season despite conceding a first-innings lead of 98.
They responded by bowling Sussex out for just 154 to set themselves a target of 253 and then openers Alex Davies and Keaton Jennings got their second innings off to a flying start with a stand of 102 in 20 overs.
Davies led the way with a 59-ball 73, with 10 fours and a pulled six over square leg off Henry Crocombe before he was bowled by Stuart Meaker.
And Jack Carson gave his side a lifeline with the late wicket of former Sussex player Luke Wells, who was lbw padding up.
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