Morning rain prevented any play until 12.40pm, with Hampshire opting for an uncontested toss before bowling Yorkshire out for 184.
Medium pacers Gareth Berg and Ian Holland struck twice apiece before tea as Yorkshire slipped to 110 for six, but Fidel Edwards was their standout overall with four for 83 from 16 overs.
The visitors later responded with 79 for five from 23 overs, including two wickets each for Jack Brooks and Tim Bresnan.
Both sides came into this penultimate round clash not mathematically clear of relegation danger.
But bottom side Worcestershire’s perilous position in their clash with Essex at Chelmsford has eased their fears, particularly Hampshire’s.
If Worcester go on to lose at Essex, they will be relegated, leaving Hampshire safe and Lancashire as Yorkshire’s remaining target. It needs nine points to eliminate them.
The afternoon session here was a helter-skelter affair which saw Yorkshire score at 4.4 runs per over in the first 15 (66 for two), two catches put down and a flurry of wickets.
Yorkshire were hurt by losing three wickets for two runs as they slipped from 70 for two in the 20th over to 72 for five in the 23rd, with Berg claiming two of them in breezy conditions.
Gary Ballance was trapped lbw playing to leg to end a fluent 30, an innings high-score.
The left-hander passed the 10,000 runs in his first-class career in the process.
Jeet Raval and Harry Brook were bowled by new ball pair Kyle Abbott and Edwards as Yorkshire slipped to 22 for two in the eighth over before Ballance and Adam Lyth (23) steadied with a stand of 48.
Brook’s first ball had seen him dropped at first slip by Holland off Abbott, with the same happening to Bresnan later on off Holland’s bowling as Tom Alsop failed to cling on behind the stumps.
Ballance was the first of those aforementioned three wickets in a mini collapse.
Berg also had Tom Kohler-Cadmore bowled for two via an under-edge as he attempted to leave alone before Holland had Lyth caught at gully as he miscued a drive.
Then, in the final over of the afternoon, the 36th, Holland bowled Jonny Tattersall as the score slipped to 110 for six.
Edwards then added three more wickets in the first half of the evening as Yorkshire slipped to 160 for nine in the 50th over.
He had Matthew Waite caught behind, Bresnan (25) yorked with a pacy out-swinger and Steve Patterson also bowled. Abbott wrapped up the innings by bowling Brooks.
Brooks then impressed with the ball to drag Yorkshire back into the game as Hampshire slipped to 68 for five in the 20th over of their reply.
Either side of Coad getting Oliver Soames caught behind, Brooks trapped Joe Weatherley lbw and Jimmy Adams superbly caught at third slip by Brook.
Brook later claimed a second catch after Bresnan found the edge off Sam Northeast’s bat, leaving Hampshire at 63 for four in the 16th. That became 68 for five four overs later when Bresnan trapped James Vince lbw.
Meanwhile, Yorkshire handed a first-team cap to new ball seamer Coad prior to play, with Coad having taken 91 Championship wickets in the first 20 appearances of his fledgling career prior to this fixture.
Coad, having sporadically played first-team cricket since debuting in 2013, finished a breakthrough summer of 2017 with 50 wickets in the County Championship and scooped three end of season awards, including Members’ and Players’ Player of the Year.
“It’s a massive honour to receive this,” said Coad, shortly after the presentation took place. “I wasn’t expecting it but apparently the family knew beforehand and didn’t tell me. I’m very honoured and privileged to have received it.
“I’ve always wanted to be capped but I didn’t expect to ever achieve it until the last couple of years. To get it now is massively pleasing.
“It’s a dream come true, I’ve always wanted to try and get to this stage of my life and the names you see on that board, just to join them is incredible and nobody can ever take it off me.
“It’s a huge milestone and you could say it’s a box ticked but we all know what the next one is now if we can get there; to try and get an England place. But to get this is massive for me and something I never thought I would achieve. This fixture last year was my breakthrough game in a breakthrough year. To be back here again a year-and-a-half later with a first-team cap is amazing.”