Wilcox initially took over on a caretaker basis in November 2013 after the dismissal of Brian Laws, before taking the job full time a month later.
The 50-year-old embarked on a record run of 28 matches unbeaten from the start of a managerial tenure, a stretch that helped Scunthorpe clinch promotion from League Two.
However, Scunthorpe have struggled with life in the third tier and find themselves in 23rd having taken just seven points from 11 matches.
Saturday's 2-1 defeat at home to Doncaster Rovers proved Wilcox's final game in charge and centre-back Dawson, who re-joined the club in May 2013, will take charge at Gillingham on Saturday.
"Scunthorpe United have today terminated the contracts of manager Russ Wilcox and assistant manager John Schofield with immediate effect," Scunthorpe wrote on their official Twitter account.
"The club would like to place on record their thanks to Russ and John for their efforts but in particular to Russ for leading the Iron into League One.
"Andy Dawson will take charge of Saturday's game at Gillingham, assisted by Tony Daws."
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