Cotterill left his position at Portsmouth to take the reins on Friday afternoon, but it was existing assistant manager Rob Kelly who led affairs from the visiting dugout and was unable to prevent a sixth defeat in seven league matches as Forest slumped into the Championship's bottom two.
Lukas Jutkiewicz headed the only goal of the game for the Sky Blues in the 57th minute, just moments after Lee Camp had made a sublime save to deny Richard Keogh from close range.
It was enough to secure only Coventry's second win of the season, although their celebrations were muted due to concern over goalkeeper Joe Murphy, who was carried off on a stretcher after lengthy treatment, resulting in 10 minutes of stoppage time.
Kelly, keeping faith with a 4-3-3 system, made two changes from the starting XI beaten 3-1 by Birmingham two weeks ago in Steve McClaren's final game in charge, Jonathan Greening and Chris Gunter coming in.
Despite the managerial focus being on Forest, Coventry boss Andy Thorn was himself forced to release a statement on Thursday confirming his commitment to City following "unfounded and malicious" rumours that he was set to quit.
But he would have been less than impressed with his side's start to proceedings.
Goalkeeper Murphy parried away Marcus Tudgay's early strike before getting down well to deny Ishmael Miller after 17 minutes as Forest posed much the larger threat.
The Sky Blues finally threatened for the first time just before the half-hour mark as Clive Platt's looping header forced Camp to tip over the crossbar.
David Bell failed to take advantage of some sloppy Forest defending when he fired across the face of goal as Coventry finished the first half stronger than they started.
The second half continued to be a lacklustre affair until it suddenly burst into life 10 minutes in.
Forest's players felt the ball had crossed the touchline for a throw-in before Wes Morgan was forced to concede a corner, and from the resulting set-piece Camp produced a stunning save to keep out Keogh's header.
But Camp was powerless as another ball quickly came back into the area and Jutkiewicz was left unmarked to head home.
Clint Hill then headed Conor Thomas's effort off the line before Camp parried Bell's strike to safety.
With 20 minutes remaining, City keeper Murphy collided in the air with Miller and, after receiving lengthy treatment, was carried off on a stretcher to be replaced by Chris Dunn.
Forest substitute Robbie Findley nodded a good chance over from close range before Jutkiewicz headed onto the top of crossbar.
And despite 10 minutes of injury time, Forest were unable to find an equaliser.