Colin Calderwood's side made it two wins in two in the league to ease the pressure on the Reds manager following an uncertain start to the campaign.
And the 4-0 victory was largely secured by Agogo, who completed his treble from close range in the 73rd minute following a pair of headers two minutes before half-time and in the 62nd minute.
Substitute Emile Sinclair added the fourth late in the game.
Gillingham came to the City Ground with a game-plan to frustrate Forest - packing their midfield and leaving just Delroy Facey up front - and for much of the first half it worked very successfully as the home side were limited to a handful of chances.
Nathan Tyson had come closest, sending a great headed opportunity straight into the arms of keeper Derek Stillie and also later being denied when his 20-yard shot bounced back off the foot of the post.
Agogo had himself already failed to make the most of a decent opportunity after a Julian Bennett cross picked him out from the left with the striker glancing a header wide.
But the Ghana international made no mistake when presented with another chance in the 43rd minute as he this time dispatched Grant Holt's cross firmly inside the far post with a thumping header.
Prior to that, Gillingham's only effort on goal in the first half was a long-range effort from veteran Steve Lomas which did little to trouble Paul Smith as it flew wide of the post.
Forest could have doubled their advantage five minutes after the break when Luke Chambers attacked a Sammy Clingan corner powerfully at the far-post, but saw Simon King clear off the line with Stillie beaten.
Two minutes later, and from another corner, Chambers was denied again a headed effort was scrambled off the line.
Gillingham were hanging on by the skin of their teeth with Wes Morgan denied in acrobatic fashion by Stillie and Agogo firing over in incredible fashion after Tyson had hit the post again.
But Agogo calmed Forest's nerves within a minute as he netted his second of the game after heading home inside the near post from Clingan's cross.
And he secured the points when he provided a simple close-range finish at the far post to complete some good work from Tyson down the left.
Substitute Sinclair increased Gillingham's misery as he made it 4-0 in the 84th minute.