Weymouth boss Hill gained a fine reputation for previously guiding Dagenham into the fourth round of the competition and he almost pulled off another shock, as his side threatened to bomb Forest out.
Gary Megson's side had looked comfortable in the first half, after taking the lead through a Gary Holt header. But, after the break, it was the visitors who could have taken the spoils.
Midfielder Andy Harris headed them level in front of 2,000 travelling Weymouth fans in the 56th minute and Kirk Jackson should have ensured there was no need for a replay, but planted what could have been a winning goal for the visitors into the side netting.
Hill had challenged his side to have a real go at Forest and make life difficult for their League One opposition - and they did exactly that.
Although Forest were on top for long periods, Weymouth's defensive resolve ensured chances were at a premium.
Holt shot badly over when presented with a chance on the edge of the box, when Jack Lester cleverly cut the ball back from the edge of the six-yard box.
Lester was then denied by a fine save from Jason Matthews, as the keeper reacted quickly to fling himself low to his left, after the striker had left two defenders in his wake before striking a low shot towards the near post.
James Perch could have done better with an attempted volley, from around ten yards out, after Eugen Bopp had lofted in a cross from the left - but he completely failed to make contact with the ball.
But Forest finally made the breakthrough in the 39th minute, following another short spell of pressure. Nicky Southall fired in a dangerous cross from the right and Holt ghosted in at the far post, unmarked, to head powerfully home from the edge of the six-yard box.
Weymouth had demonstrated that they were a threat on the break themselves, but Chukki Eribenne was denied a shooting chance when Wes Morgan produced a fine block tackle to thwart the striker.
And, after a half-time team talk from Hill, the Nationwide Conference South side seemed revived. They began with pace and produced some slick football - which eventually led to an equaliser.
A corner from Lee Elam left keeper Rune Pedersen needing treatment, after he had punched clear amid a melee of players. But, from the second flag kick, Elam swung the ball dangerously to the far post, where Harris powered a header home in a similar manner to Holt's opening goal for Forest.
Southall came close to putting Forest back ahead immediately, but saw Matthews pull off another outstanding save.
And it was the visitors who should have netted again, Steven Clark fired in a superb cross from the right to pick out Jackson's run to the far post, where the striker headed a fine chance into the side netting.