A goal either side of half-time from Grabban, which took his tally for the season to 19, put daylight between Forest and their struggling visitors and Yates put the icing on the cake with a near-post finish five minutes from time.
Juninho Bacuna was shown a straight red card late on for a dangerous lunge on Nuno Da Costa before Karlan Grant scored a penalty in the final minute of six minutes of stoppage time after Joe Worrall shoved Fraizer Campbell to the floor inside the box.
Sabri Lamouchi made one change to the side that drew 1-1 at Sheffield Wednesday last weekend as Ben Watson came back in to captain the side in place of Yates who dropped to the bench.
The Reds should have been in front inside four minutes. Joe Lolley went on a strong run down the right-hand channel and pulled the ball back into the penalty area for Sammy Ameobi but having picked his spot, the winger crashed his effort against the base of the post.
Brice Samba was called into action on 15 minutes as he was forced to push behind a curling free-kick from Alex Pritchard from the edge of the penalty area before two resulting corners from Huddersfield caused problems with Forest at full-stretch to clear the danger.
The visitors were enjoying the better of the exchanges midway through the first half. A clever free-kick routine saw Karlan Grant drill an effort towards goal from inside the box but Ameobi did well to turn it the ball over the crossbar from inside his own six-yard box.
Two minutes before the interval and Forest had the breakthrough. A deep corner from the right drifted all the way out to Joe Worrall in the left-hand channel, he delivered an excellent ball back into the penalty area and an unmarked Grabban produced a thunderous first-time volley to beat Jonas Lossl at his near post.
Yates replaced Brennan Johnson at the interval and just 50 seconds into the second half, Forest doubled their advantage. Ameobi slipped a ball through behind the Huddersfield back four and Grabban let it run in front of him before he slid a low strike under the body of Lossl into the back of The Terriers' net.
Huddersfield almost found an instant reply as a corner caused all sorts of problems inside the Forest penalty area but Ameobi's header bounced back off his own crossbar and The Reds were able to clear the ball to safety to preserve their two-goal lead.
There was an uncharacteristic miss from Grabban as he attempted to seal his hat-trick. Ameobi was the provider again with a superb pass into the right-hand side of the penalty area but Grabban's low drive went wide of the far post with Lossl stranded.
Samba hadn't really been required in the second half but he was called upon to make a strong save with 15 minutes to play. Substitute Juninho Bacuna struck a powerful free-kick towards goal from the edge of the box and the Forest number one made an excellent one-handed stop to palm the ball away.
As Huddersfield were pushing for a goal at one end, Forest were looking to extend their lead at the other and they sealed the points with five minutes to play. Yates got across his man at the near post and was in an excellent position to stick a foot out and send Lolley's corner into the net from close range.
Huddersfield saw their afternoon brought to a disappointing end courtesy of Bacuna's late red card. Referee David Webb had already awarded Huddersfield a free-kick before Bacuna hacked Nuno Da Costa down with a poor challenge on the halfway line and it was enough for Webb to show the former FC Groningen man a straight red.
The visitors did grab a consolation goal in the final minute of stoppage time after Worrall shoved Campbell to the ground inside the penalty area. Grant stepped up to thump the ball beyond Samba into the corner of the net to give them a late goal to take back to West Yorkshire.
Forest: Samba; Cash, Figueiredo, Worrall, Ribeiro; Watson (c) (Mighten 90'), Silva (Carvalho 73'); Lolley (Bostock 90'), Johnson (Yates 46'), Ameobi (Da Costa 73'); Grabban
Subs not used: Smith, Jenkinson, Dawson, Walker
Goals: Grabban 43', 46', Yates 85'
Bookings: Watson 44', Ameobi 62'
Huddersfield: Lössl; Duhaney (Bacuna 74'), Stearman (Stankovic 70'), Schindler (c), Toffolo; Hogg (Chalobah 60'), O'Brien, Willock (Campbell 70'), King, Pritchard (Smith Rowe 60'); Grant
Subs not used: Coleman, Kachunga, Quaner, Daly
Goal: Grant (pen) 90'
Sent off: Bacuna 89'
Referee: David Webb
