Blackburn left it late to avoid a shock home defeat against Coventry, while Middlesbrough were held 1-1 by West Ham in Saturday's only all-Premier League match.
Fulham were pegged back by Swansea at the Liberty Stadium and Sheffield United and Hull City will also have to play again after a 1-1 draw.
Chelsea were forced to come from behind against Watford at Vicarage Road as they made a positive start to life without Luiz Felipe Scolari.
New manager Guus Hiddink, sitting alongside Roman Abramovich in the stands, did not see any goals in the first half as Watford withstood an onslaught from the Blues.
The Hornets then took a surprise lead in the 69th minute when substitute Tamas Priskin raced clear to loft the ball past Petr Cech.
A huge upset looked to be on the cards but Nicolas Anelka had other ideas and hit a quick-fire hat-trick as Chelsea became the only team to progress to the last eight on Saturday.
The Frenchman's improvised hook brought Chelsea back on level terms after 75 minutes and a neat header from Ashley Cole's cross made it 2-1.
Cech had to be alert to keep out Jobi McAnuff before Anelka grabbed his third in stoppage time.
Blackburn were thumped 4-1 by Coventry in the third round of last year's competition and it looked like history would repeat itself until Christopher Samba's last-gasp goal rescued a 2-2 draw.
Coventry fell behind after just two minutes as Roque Santa Cruz showed the sort of finishing prowess that made him a big-money target for Manchester City in the transfer window, rifling home a volley from just inside the area.
However, Chris Coleman's Sky Blues improved as the match went on and equalised just after the hour mark courtesy of Aron Gunnarsson's fabulous strike from distance.
Things got even better for the visitors 15 minutes later when Michael Doyle rounded off a flowing move by sending a deflected shot beyond Paul Robinson.
But they were denied a place in the quarter-finals as Samba took advantage of some hesitant defending to score from 12 yards.
Swansea, flying high in the Championship and on a long unbeaten run, were tipped to cause an upset against a Fulham side that has been prone to travel sickness in the Premier League this season.
The Swans made a bright start but Fulham took the lead on the stroke of half-time when Garry Monk inadvertently turned the ball into his own net following a mistake by goalkeeper Dorus de Vries.
Fulham's advantage did not last long as Jason Scotland produced an exquisite piece of skill in the 52nd minute to make it 1-1. The Trinidad & Tobago international turned Aaron Hughes on the edge of the box before firing a low shot past Mark Schwarzer.
Swansea continued to push forward in the closing stages but could not force a winner and they will now resume battle at Craven Cottage.
Middlesbrough came within seven minutes of a confidence-boosting victory over West Ham but they will now hope to finish the job at The Riverside.
Stewart Downing broke the deadlock for Boro in the 22nd minute, poking the ball in from close range after West Ham had failed to clear a teasing cross from Gary O'Neil.
Gareth Southgate's side stayed in front until deep into the second half but they could not cling on as Herita Ilunga headed home Savio's free-kick.
Sheffield United and Hull both kept their FA Cup dreams alive after playing out a 1-1 draw at Bramall Lane.
The Tigers kept Chelsea at bay for 90 minutes last weekend but were stunned after just seven minutes as Greg Halford headed home David Cotterill's cross.
Hull drew level in the 34th minute as Andy Dawson's free-kick was nodded in by Kamil Zayatte, but neither side was able to grab a winner in a pulsating second half.