Steve Cooper's battling side earned the early advantage when Brazilian defender Renan Lodi lofted a ball into the Chelsea area in the 13th minute.
Goalkeeper Edouard Mendy, who had not played since suffering an injury playing for Senegal at the World Cup last year, came off his line but misjudged the ball's flight and Awoniyi rose between two defenders to head home.
Raheem Sterling, lively from the start of the match, scored Chelsea's 51st-minute equaliser with a simple close-range shot from a pinpoint cross by defender Trevoh Chalobah.
The goal gave the home side the confidence that has been so lacking in recent games.
Conor Gallagher, Joao Felix and Enzo Fernandez all had second-half efforts saved or blocked and Sterling doubled his tally in the 58th minute with a fine curling shot after a surging run and pass from Ruben Loftus-Cheek.
Chelsea, who have dropped into the second half of the table after a string of defeats, looked to be heading for their first home Premier League victory since a 1-0 win over Leeds on March 4.But barely four minutes later Forest were back level, Awoniyi grabbing his second with a close-range header after a long throw.
"The most important thing was to get the goals and that brought us into the game," Awoniyi told Premier League Productions.
"Many times we sat off but that's the game plan and that's what we did." He said that as a striker he wanted to score every game but he could not achieve that without his teammates.
"If they don't bring the ball into the box, there is no way I can get the chance to score so it is all teamwork," he added. "We still have two finals to go, the next one is Arsenal and from there we will see what happens."
The result kept the visitors three points clear of the drop zone with two games of the season left and their travelling supporters, in fine voice throughout the game, celebrated as though they had won.
Chelsea, under caretaker manager Frank Lampard, remained 11th as they limp towards the end of a disappointing campaign.