Darren Huckerby netted his second successive hat-trick, having also struck a treble in Coventry City's 7-0 FA Cup win over Macclesfield Town a week earlier.
Huckerby scored just before and after half-time against Forest. Paul Telfer made it 3-0 before Huckerby registered again.
Coventry's first win in eight Premier League matches left them 17th with 20 points, three points above the relegation places. Forest remained bottom with 13 points.
Southampton beat Liverpool at home in the Premier League for the first time in nearly eight years.
The Saints took a 63rd-minute lead with a James Beattie penalty, and eight minutes later it was 2-0 as John Arne Riise scored an own goal.
Southampton climbed to 12th with 25 points, while Liverpool failed to capitalise on their chance to join Leeds United at the top of the table.