The result leaves the Gunners seven points clear at the top, though that margin could drop to four if Aston Villa beat Everton on Sunday.
The London side dominated possession throughout the match but repeatedly failed to convert chances, as Forest defended with a disciplined, well-drilled structure.
Arsenal's key opportunities fell to Gabriel Martinelli, who missed a point-blank effort from a deflected Noni Madueke cross and steered another big chance wide before half-time.
Mikel Merino also headed a Rice free-kick narrowly over the bar, while Gabriel Jesus and Bukayo Saka were denied by acrobatic saves.
The standout performer was Forest goalkeeper Matz Sels, the Belgian recalled after earlier struggles this season. Sels produced several crucial stops, including a spectacular reflex save from Saka, a smothered close-range flick from Jesus, and a volley from Declan Rice in the 59th minute.
His performance underlined his importance to Sean Dyche's side and justified his return to the starting line-up following injuries to John Victor.
Forest, 17th in the table and five points above the relegation zone, matched the defensive solidity expected under Dyche, with defenders and goalkeeper working cohesively to repel Arsenal's attacks.
The visitors also had a VAR incident checked when Ola Aina appeared to handle a cross, but it was deemed a natural action, leaving Arsenal manager Mikel Arteta visibly frustrated.
Arteta lamented the lack of a “magic moment” and expressed hope that Everton, managed by his former boss David Moyes, could impact the title race.
Mikel Arteta has said Arsenal's consistency this season is convincing his players they can achieve something historic, with the club still competing for four trophies as the campaign reaches its decisive stage.
Mikel Arteta played down distractions, brushed aside injury fears and insisted Arsenal's full focus was fixed on Nottingham Forest as the Gunners prepared for another demanding Premier League away game.
Tijjani Reijnders insists Manchester City's confidence in the Premier League title race remains strong as attention turns to Saturday's Manchester derby at Old Trafford.
Liam Rosenior has said Chelsea were "still in this tie" after a dramatic 3-2 defeat to Arsenal in the Carabao Cup semi-final first leg on Wednesday night.
