But Guimaraes secured the three points for the visitors with a brilliant finish in the 66th minute.
Victory lifts Newcastle up to seventh in the table, closer to the European places.
"We have to get Europe again," Guimaraes told Sky Sports. "Try for Champions League, I know it's difficult but definitely for Europa League."
Newcastle finished fourth in the Premier League last season to secure a place in the Champions League for the first time since 2003.
The Magpies are 11 points off fourth-placed Tottenham, but Kieran Tripper said Newcastle have a "never say never" attitude towards qualifying for Europe's premier competition.
Meanwhile, a defeat for Forest leaves them just two points clear of the relegation zone.
After a dramatic 4-4 draw against Luton Town the previous weekend, Eddie Howe's side played out another thrilling game.
Newcastle pulled off a clever short corner routine to go ahead in the 10th minute, with Kieran Tripper whipping a fierce ball to the back post where Guimaraes had timed his arrival to perfection to poke home the volley.
The atmosphere around the City Ground improved drastically when Morgan Gibbs-White threaded a well-weighted ball through the centre of the pitch and Elanga outpaced Newcastle's defenders to slot it past Martin Dubravka.
In a first half full of momentum shifts, Newcastle restored their lead as Schar met Sven Botman's header on the half-volley, firing it back across goal and into the bottom left corner.
Again Forest responded, with Hudson-Odoi's strike from 20 yards out taking a big deflection off Lewis Miley to wrong-foot Dubravka.
Nuno Espirito Santo's side may feel hard done by after referee Anthony Taylor waved away two penalty appeals, first for a light shove on Elanga by Anthony Gordon and then for Dubravka's outstretched arms bringing down Taiwo Awoniyi.
Forest pushed forward in search of a third equaliser, but a lack of a clinical finisher in front of goal cost them as they failed to make a single shot on target in the second half.
Guimaraes shines to save Newcastle
One moment of brilliance from Guimaraes proved the difference for Newcastle and he was met with a huge roar from the away fans after full-time.
The Brazilian made a clever run into the box for his first, and he showed the same footballing intelligence for the winning goal - latching on to a loose pass from Elanga before looking up and perfectly picking out his spot in the Forest net.
However, concerns will remain for Howe over his side's defensive record, having failed to keep a clean sheet since before Christmas.
The Newcastle boss was forced to defend Dan Burn earlier in the week, calling him "Newcastle through and through" after he came under scrutiny for his performance against Luton.
Aston Villa's Leon Bailey and Luton's Chiedozie Ogbene tormented the defender for speed in Newcastle's last two league outings and Elanga was able to target the same weakness here, peeling away from Burn for Forest's first equaliser.
Forest still searching for first league win of 2024
Forest find themselves in a perilous position in the Premier League, sitting 16th in the table with only a two-point safety cushion.
They may feel unlucky to have come away from Saturday's match without a point, having twice fought back from a goal down and pushing hard in the dying moments.
Perhaps Wednesday's FA Cup fourth-round replay took it's toll, with Forest needing 120 minutes plus a penalty shootout to overcome Championship side Bristol City.
Both of Forest's two victories in 2024 have come in FA Cup replays, but they are still without a much-needed league win.
Nuno will be boosted by the imminent return of Ivory Coast trio Serge Aurier, Willy Boly and Ibrahim Sangare along with and Nigeria's Ola Aina, who are preparing to face one another in the Africa Cup of Nations final on Sunday.