The Reds were guaranteed a top-six finish with victory or by bettering eighth-placed Blackpool's result against already-promoted Newcastle - and they did both.
There was little to get excited about in the first half but the second period was barely a minute old when Forest opened the scoring through Chambers. Moussi extended the lead 10 minutes later before Earnshaw netted his 14th goal of the season after 76 minutes.
Blackpool's 4-1 defeat at St James' Park means Forest will definitely be one of the four teams in the play-offs as the Reds' outside hopes of winning automatic promotion were finally extinguished by second-placed West Brom's 3-2 victory at Doncaster.
Paul Anderson returned to the Forest team after missing the Easter fixtures with an Achilles tendon problem. He replaced Radoslaw Majewski, who dropped down to the bench, in the only change made by Reds boss Billy Davies. Ipswich were unchanged.
The only goalmouth action of any note in the first half saw Earnshaw and Moussi shoot wide from 12 yards out, while Connor Wickham did likewise for the visitors.
Immediately after the restart, Forest went ahead. The goal was credited to Chambers but the defender knew little about it.
Earnshaw's shot from the edge of the box struck him eight yards out and left Ipswich goalkeeper Brian Murphy totally wrong-footed as the ball drifted into the bottom right-hand corner.
Moussi made it 2-0 in the 57th minute when he scrambled the ball over the line following Chris Gunter's near-post cross.
Wickham dragged another shot wide just before the midway point of the second half.
Earnshaw scored from close range to make sure of an incredible 16th win in the last 18 home league games for Forest, who last lost at the City Ground in September.
Daryl Murphy and Jonathan Walters shot over the crossbar before Forest goalkeeper Lee Camp was forced into his first piece of meaningful action in the first minute of injury-time.
David Norris' volley looked like it might creep in but Camp tipped the ball over the crossbar.