Tottenham haven’t lifted a major trophy since 2008 when current Middlesbrough manager Jonathan Woodgate scored the League Cup winner against Chelsea.
Woodgate was on course to dent Tottenham’s bid to end that wretched run when Ashley Fletcher put Middlesbrough ahead in the second half at the Riverside Stadium.
But Brazilian winger Lucas headed the equaliser to send the tie to a replay in north London later in January.
Keen to win silverware in his first season, Tottenham boss Jose Mourinho had made just three changes, with Son Heung-min returning from a three-match suspension and replacing the injured Harry Kane.
Son twice came close to putting Tottenham ahead in the first half and the visitors should have gone in front when Ryan Sessegnon fluffed his chance from Serge Aurier’s cross.
Tottenham were never completely convincing and, with only one clean-sheet since Mourinho took over, there was a familiar feeling when Middlesbrough snatched the lead in the 50th minute.
Fletcher beat Tottenham’s attempt to play offside, ran through on goal and fired past Paulo Gazzaniga, with no VAR in use to question the goal.
Tottenham hit back in the 61st minute, Aurier’s cross picking out Lucas for a thumping header.
At Stamford Bridge, there were signs that Hudson-Odoi might be emerging from his sophomore slump.
Hudson-Odoi has endured a difficult season as injuries and poor form have ruined the positive impact he had last season.
But the 19-year-old had made his name with a breakthrough display against Forest in the FA Cup last year.
So perhaps it wasn’t a surprise that Hudson-Odoi was more like his old self as he opened the scoring against the second tier side in the sixth minute.
Taking possession on the right, the England winger cut inside and fired into the bottom right corner for just his second goal of the season and his first since September.
Blues boss Frank Lampard had made nine changes, but Chelsea — wearing a retro home kit in tribute to their first FA Cup final win against Leeds in 1970 — doubled their lead in the 33rd minute.
Hudson-Odoi’s shot was parried straight to Ross Barkley and the midfielder poked home at the back post.
Crystal Palace became the third Premier League club to be knocked out this weekend as second tier Derby won 1-0 at Selhurst Park.
Palace boss Roy Hodgson made five changes — a relatively small number compared to most other Premier League clubs this weekend — but the Eagles still went the same way as Aston Villa and Brighton, who bowed out against Championship opposition on Saturday.
Wayne Rooney was in the Derby attack and his side took the lead in the 32nd minute when Jayden Bogle fizzed a cross to the far post and Chris Martin tapped into the net.
In a VAR rarity, referee Michael Oliver used the pitch-side monitor to change a decision, sending off Luka Milivojevic after initially booking the Palace midfielder for kicking out at Tom Huddlestone in the 63rd minute.
Sheffield United avoided a non-league upset with a 2-1 win against AFC Fylde at Bramall Lane.
United boss Chris Wilder made 11 changes to his Premier League side, but goals from Callum Robinson and Leon Clarke put them in control.
Fylde, who play in the fifth tier National League, got a consolation goal from Jordan Williams in the 78th minute.