Hudson-Odoi earns Forest draw at Bournemouth
Last updated on .From the section Premier League
But Hudson-Odoi equalised just before the break, beating Ryan Christie and Illia Zabarnyi before bending a right-footed effort into the bottom corner.
Bournemouth ended with 10 men as Philip Billing was shown a straight red in the 83rd minute for fouling Hudson-Odoi.
The draw means Bournemouth are up to 12th in the table while Forest move a point and a place above Luton into 16th.
The Cherries started the match brightly, winning two corners in the opening five minutes and scoring from the second one through Kluivert's close-range finish.
Despite going behind so early in the contest, Forest stayed in the game and defended doggedly.
Taiwo Awoniyi, in for the injured Chris Wood, was unable to take his chances as Nuno Espirito Santo's side looked to get back on level terms.
He lost his footing at the vital moment after being played in by Morgan Gibbs-White and then had a weak effort saved by Bournemouth goalkeeper Neto.
Former Chelsea winger Hudson-Odoi equalised with a classy strike at the end of the first half, but neither side could find a winner after the break with few clear-cut chances created.
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The stop-start nature of the game played into Forest's hands with Bournemouth unable to capitalise on their fast start.
Dominic Solanke made his 200th Cherries appearance but was unable to mark the milestone with a goal as the visitors' defence kept him quiet for the majority of the contest.
Andoni Iraola's side had nine shots but just two on target as they struggled to hit their free-flowing best - and will see this as a missed opportunity to end a four-game winless run in the top flight.
They are next in league action against Fulham on 10 February (15:00 GMT kick-off).
Forest had lost their past two league games against Brentford and Arsenal but an improved display secured them a point against Bournemouth which helps their survival hopes.
Hudson-Odoi's second league goal of the season cancelled out Kluivert's early opener and the visitors defended stoutly to restrict Bournemouth to few chances.
Goalkeeper Matz Sels also put in a solid performance on his Forest debut after signing from Strasbourg on deadline day, dealing well with the hosts' set pieces throughout.
Forest are at home to Newcastle in their next league match on 10 February (17:30 kick-off).