The Clarets looked to be controlling the game but deadline day signing Hudson-Odoi had other ideas, curling a stunning strike into the top corner on the hour mark.
Burnley had taken a first-half lead through Zeki Amdouni's well-taken shot into the bottom corner after fine work by the lively teenager Luca Koleosho.
Swiss striker Amdouni, who scored the winner here in the Carabao Cup just 20 days ago, had already given goalkeeper Matt Turner a warning sign with a stinging drive from range before netting a cool finish.
Burnley's night then ended on a sour note when top scorer Foster was sent off for violent conduct in injury time, after a needless off-the-ball elbow on Ryan Yates.
Following three straight defeats, the result allows Burnley to get off the mark this season, while Forest have made a decent start to the campaign with seven points from five games.
A point apiece a fair result
These two sides met at the City Ground for the second time in three weeks and it seemed like a repeat result was on the cards.
Burnley's club record signing Amdouni converted a 90th-minute winner to send his side through to the EFL Cup third round at the expense of Forest last month.
And he netted again on Monday night with a drilled effort from the edge of the box after being set up by Italy youth international Koleosho, who outpaced home skipper Joe Worrall close to the byeline.
Koleosho impressed, giving Argentine World Cup winner Gonzalo Montiel the runaround throughout the game before being replaced by Belgium international Mike Tresor.
This was a solid showing from Kompany's men in their first away game of the campaign after conceding three or more goals in all their matches so far.
A club-record five consecutive Premier League defeats was avoided but their return to the top-flight has been a struggle after cantering to the Championship title last season.
Forest, though, hit back in the second period through Hudson-Odoi's brilliant finish with the side highlighting their prowess in front of goal by scoring in their 12th straight top-flight game.
The equaliser came from Taiwo Awoniyi's knockdown and the Nigeria striker is becoming an integral part of a Forest side that keeps changing personnel, now contributing a goal or assist in his last nine league games.
The two sides face Manchester clubs on Saturday - Forest travelling to Treble winners and league leaders Manchester City (kick-off 15:00 BST), while Burnley host United on the same day (20:00).