The striker came close moments later when his back-post header went wide from Adam Hammill's cross.
In an unremarkable second half, Eggert Jonsson drew a good save from Colin Doyle with a fizzing 25-yard effort.
The Championship side went ahead with 15 minutes to play when Elliot converted at the second time of asking when his initial shot had hit the post.
Jean Beuasejour came close to doubling his side's lead in the 85th minute, but his side-footed drive was palmed away by De Vries.
Substitute Steve Fletcher nearly found an equaliser in the dying minutes, only to see his shot smothered by Doyle. Stephen Hunt almost put away the rebound, but the Irishman fired over from close range.