The 28-year-old Nigeria international opened the scoring at Stamford Bridge after just two minutes.
Igor Jesus added a second from the penalty spot with 15 minutes played and Matz Sels denied Cole Palmer from 12 yards in first-half stoppage time to preserve the two-goal lead at the break.
Awoniyi then bagged his second of the night with 52 minutes on the clock while Joao Pedro scored a late consolation goal for the Blues in the third minute of stoppage time at the end of the game.
The result eased Forest's relegation concerns by lifting them on 42 points in 16th place as Chelsea sit outside the European places in ninth place, with 48 points.
"It's an important win, credit to the players, to the coach and the staff. We came here with the mentality of getting three points, which we need, and I'm so pleased that we managed to win," the player told his club's website.
"It was an amazing start and we had control of the ball in the first five minutes, which is what the coach wanted from us.
"It was an amazing delivery and I was in the right place at the right time to score. It was a team effort, not just about the goals. Morato as well, and the players that came on from the bench. It's about the team and being ready for the team.
"Next up we have [Aston] Villa, which is another important game. We're taking each game as it comes and going into it with a three-point mentality.
"I believe we'll go there with that mentality against Villa, and we will see what comes next until the end of the season."