Greg Cunningham headed the hosts in front after seven minutes, climbing well at the far post to nod home Scott Wagstaff's cross from the right.
By the 13th minute it was 2-0 as Emmanuel-Thomas beat a defender inside the box and crossed low for Sam Baldock to net his 12th goal of the campaign from close range.
City goalkeeper Ellliott Parish preserved the lead with fine saves to deny Luke Freeman, Jordan Burrow.and Roarie Deacon, while the dangerous Freeman saw a shot strike a post.
Emmanuel-Thomas then settled matters by converting a low Baldock cross after 69 minutes and then firing in a spectacular fourth after a strong solo run three minutes later.
Francois Zoko netted a consolation goal for Stevenage from a Freeman cross in the 74th minute.
In the closing stages City sent on veteran defender Louis Carey for his 646th appearance for the Robins, breaking an all-time club record held by John Atyeo.