Worcestershire 242 (Coad 5-33) and 22 for 0 trail Yorkshire 407 (Bean 135, Lyth 79, Hill 53, Finch 5-100) by 143 runs
Coad finished with 5 for 33 from 21 overs - the 11th five wicket haul of his career - and he was well supported by Mark Steketee and Matthew Fisher as Worcestershire were forced to follow on 165 behind.
But Finch and Gibbon ate up invaluable time to bolster Worcestershire's hopes of securing a draw as they batted for 35.3 overs during a last wicket stand of 63 - the highest partnership of the innings.
It kept Yorkshire out in the field for two hours and put extra miles into the legs of some of their bowlers when they were itching to have a crack at the Worcestershire top order for a second time.
Gibbon's 41 not out was a career best and Finch followed up his five-wicket haul yesterday by making a defiant 24 before he was last out.
Yorkshire are probably sick of the sight of Finch as his three sixes in the final over of the Vitality Blast game at New Road in May off Fisher earned his side an unlikely victory.
Coad picked up his second five-wicket haul of the campaign after his 5 for 54 against Sussex at Hove in April and was a constant threat.
With just nine overs remaining, Finch and Gibbon went straight back out into the middle in nightwatcher mode to open rather than the established batters.
They again proved difficult to dislodge as Worcestershire reached 22 for nought, 143 in arrears heading into the final day.
Worcestershire resumed the day on 46 for 2 and Jack Haynes looked in good nick as he collected a trio of boundaries off Coad.
He square drove, cover drove and flicked him off his legs to the ropes. But Coad had his revenge with the final ball of the sixth over of the day as Haynes (29) was adjudged lbw.
Azhar Ali battled away against some probing bowling but fell victim to Coad after he switched ends. He was squared up by a delivery and provided George Hill with a comfortable catch at first slip.
Coad bowled five successive maidens before being rested after morning figures of 8-5-14-2.
Steketee came into the attack and he accounted for Ed Pollock who nicked a ball angled away to second slip.
Adam Hose needed 26 balls to get off the mark but he then started to score freely with a succession of boundaries against Matthew Fisher and George Hill.
But there was further reward for Skeketee as former Yorkshire all-rounder Matthew Waite appeared to edge a delivery onto his pad before the ball ballooned to third slip.
Hose was unbeaten on 37 when rain brought about an early lunch but he only added a single after play resumed before being beaten all ends up by Fisher and losing his off stump
Joe Leach and Josh Baker added 31 for the eighth wicket before the latter was caught at cover driving at Coad.
Leach made an accomplished 33 off 44 balls but then was undone by a ball which lifted on him and Coad held a head high catch at third slip.
Worcestershire were then 179 for nine with 47 overs still remaining in the day. But then Finch and Gibbon joined forces to blunt Yorkshire's efforts to quickly wrap up the innings.