In the first half, Forest knocked them out of their rhythm as they were off the pace.
In the 55th minute, Awoniyi scored on a rebound after his first attempt hit the post.
The hosts had several chances to stop the counter- attack.
They were indebted to keeper Dean Henderson, who made a number of saves, the best of which was in the last minute when he saved a close-range headed goal by the Dutch defender.
Forest moved to 19th place with nine points from 12 games. The Reds are in seventh place and have yet to win an away game.
After a poor start to the season, a win over Manchester City and a win over West Ham United in the space of a week gave the side some much needed confidence.
They were expected to roll over a Forest side with many injuries.
Steve Cooper's side were magnificent as they were able to get a result that could kick start their season.
The turning point has to be this one. Ryan Yates said that they can beat anyone if they beatLiverpool.
In the first half, the Reds struggled to break down Forest's defensive ranks, with Van Dijk missing a headed opportunity.
Jesse Lingard shot straight at Alisson from one break on the counter attack.
Awoniyi was a thorn in the side of the Reds and Forest's goal came from a free kick conceded by Joe Gomez.
Steve Cook hit a low cross towards Awoniyi who shinned his first attempt against the post before steering home the rebound.
Forest could have made it 2-0 on the counter-attack when Morgan White was denied by a block by James Milner and Brennan Johnson went close, but they missed out.
As the minutes wore on, Henderson was forced into action again to deny Trent Alexander-Arnold from close range and the goalkeeper was to produce more heroics.
In the final moments of the game, Van Dijk volleyed wide and then Henderson saved the ball from going in the net.
This is only a victory. "We have to build on that and keep going," Cooper said.
It's important to have good moments at both ends in order to win a game. Today we did that.
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