The 24-year-old skipper rescued his team from 24 for four in their second innings, making 82 off 137 balls, with 8 fours and a six, as they recovered to 200 all out. Ben Coad claimed four for 61 and Jack Brooks three for 44.
That set Yorkshire a testing target of 321 to win on a pitch still offering assistance to the seamers. By the close of a day interrupted three times by bad light, the visitors had reached 49 for one, with Adam Lyth unbeaten on 25, and trailed by 271.
The day began with Somerset six without loss, leading by 126. They soon plunged into trouble as Coad and Brooks made good use of the new ball to pick up four cheap wickets between them.
Coad had Marcus Trescothick caught at second slip and George Bartlett taken by wicketkeeper Andrew Hodd, while Brooks bowled Matt Renshaw and James Hildreth, both driving.
With their lead only 144, Somerset needed a partnership. It was supplied by Abell and Steve Davies, who added 66 before Davies, on 33, edged Coad to first slip where Cheteshwar Pujara took the catch.
Lewis Gregory got off the mark with a six over mid-wicket off Brooks and followed up with a four off the next delivery before Matthew Waite had him lbw playing no shot for 14.
At lunch the scoreboard read 113 for six and Somerset had extended their advantage to 233. Craig Overton contributed a bright 18 before being bowled by a full, swinging delivery from Brooks to make it 129 for seven.
Abell stood firm, playing an innings of sound technique and good judgement to reach his half-century off 103 balls, with 6 fours. Dom Bess (14) shared an important stand of 42 before driving at Waite and edging a catch to second slip.
Yorkshire gave Abell two lives, Hodd dropping a touch chance off Coad when the Somerset captain was on 24 and Jack Leaning spilling a more straightforward catch at gully off Tim Bresnan with the batsman on 69.
Abell hit a rare six, clipping Waite over backward square as Somerset’s lead passed the 300-mark. Josh Davey made a useful 11 before driving Bresnan to Gary Ballance and the innings ended when Abell was bowled by Coad, aiming a big shot.
Yorkshire were 13 without loss when bad light brought an early tea. On the resumption they had doubled that score when Harry Brook, on 15, edged a drive off Davey to Trescothick at second slip.
At 44 for one bad light intervened again and, although the players returned briefly while five more runs were added, the umpires soon took them off again. After 25 minutes of consultation, aided by a light metre, they abandoned play for the day.