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Albion were set on their way by a comic fifth-minute headed own goal from Luton captain Kal Naismith.
Goals from strikers Callum Robinson, in the 36th minute, and Karlan Grant, in the 66th minute, appeared to have got the job done before Cornick began the Hatters revival on 70 minutes.
That quick run of second-half goals came immediately after the 10-minute hold-up that followed Clark’s nasty-looking collision with Johnstone, back for his first start of the season after a delayed return from European Championship duty.
If they had done so, Albion would only have had themselves to blame for not making the game safe sooner.
Albion had enjoyed a lucky break for their early opener when Darnell Furlong’s long throw made it all the way to the edge of the six-yard box and Naismith climbed highest but succeeded only in getting a thin touch off the top of his head, which diverted the ball past his own goalkeeper Simon Sluga.
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From another Furlong throw, Sluga tipped a header over from defender Matt Clarke, who then tested him again with a near-post flick, and Grady Diangana’s header was deflected wide before Albion doubled their lead.
From Alex Mowatt’s corner, Sluga flapped under pressure from Clarke and Robinson stole in to squirt a far-post header over the line.