After a painful practice, Bairstow ruled himself out before the start of the third day, Will Luxton taking his place as a full playing substitute.
The new regulations mean Bairstow has to sit out the next game against Hampshire, but Yorkshire hope he will make the following match at home to Sussex on 24 April.
Timm van der Gugten struck early when Finlay Bean edged a catch behind to Chris Cooke, but that brought debutant Whiteman in to join Adam Lyth in a confident stand of 76 at four an over.
Whiteman, the Doncaster-born batter who has spent his first-class career with Western Australia, stroked nine boundaries in a promising debut, although he was dropped at slip off Ben Kellaway before being pinned lbw on the back foot by Crane.
The spinner’s googly then accounted for James Wharton lbw fourth ball and sub Luxton (14) missed an attempted pull to be bowled behind his legs.
The obdurate Lyth reached 37 before New South Wales paceman Ryan Hadley produced an effort delivery that the opener steered to clip.
Crane’s sharp turn was soon enough to account for George Hill, who was bowled playing down the wrong line.
Revis and substitute captain Dom Bess staged a mini-revival as Yorkshire maintained their quicker scoring rate, Bess surviving some nervy moments against Crane in his breezy 32 before being given lbw to van der Gugten.
Hadley’s shorter deliveries then accounted for Logan van Beek (6) and Revis, the ball after he hit a six to reach 50 off 78 balls.
Crane wrapped up his excellent five-for by having Ben Coad caught at slip for 12.
The spin-heavy trend continued with Bess having Asa Tribe plumb leg before for 19 for his 300th first-class wicket as Glamorgan restarted in the same painstaking mode as their first innings against an accurate attack.
A jittery Sean Dickson also fell early to Bess while Eddie Byrom made a solid 40 before holing out off van Beek, with the first three wickets all falling to rare attacking shots.
A long evening session in chilly sunshine produced few fireworks as Kiran Carlson (32 not out) and Colin Ingram (18 not out) settled for stabilising their team’s position.