Priskin's goal came completely against the run of play but Anelka equalised six minutes later from close range and before Watford could regroup he headed in a cross two minutes later.
The France striker completed his treble in stoppage time to make Chelsea the first side through to the quarter-finals.
"Guus came down to the changing room to congratulate the players," stand-in manager Ray Wilkins told reporters.
"He is thrilled with the victory and knows he has a wonderful group of players to work with," added Wilkins who is to stay on as Hiddink's number two.
The day's other ties ended all square with Premier League Blackburn Rovers and Fulham relieved to still be in the competition.
Blackburn were indebted to a last-gasp equaliser from Christopher Samba as they drew 2-2 at home to Championship side Coventry City.
Fulham survived a severe test against another team from the second tier, hanging on for a 1-1 draw at Swansea City.
In the only all-Premier League clash West Ham United drew 1-1 at home to Middlesbrough while Yorkshire rivals Hull City and Sheffield United also ended 1-1.
Watford hardly got a kick in the opening 15 minutes on an awkward, muddy Vicarage Road pitch as a strong Chelsea line-up, which included debutant Michael Mancienne at right back, stroked the ball about.
LIVELY DISPLAY
Didier Drogba had been frustrated by Scolari's reluctance to combine him with Anelka in attack but he repaid Wilkins's faith with a lively display in a central role, as Anelka started out on the right.
"I had a chat with Didier in the week, a little one to one, because I just had a gut feeling about it," Wilkins said.
"They are two exceptional players and when you see them up front together it's a pretty formidable sight. It has been an in-and-out season for Didier but it's nice to see the big man back, he's a big character."
Drogba was denied by Watford keeper Scott Loach after 20 minutes while Frank Lampard, playing against the club where his father is on the coaching staff, also fired in several fierce shots from outside the area.
Michael Ballack somehow failed to score from two metres in the second half as Watford hung on.
The home team then took the lead out of the blue when Priskin, who had only been on the field for a few minutes, sprang the offside trap and kept his nerve to lift a delicate shot over on-rushing keeper Petr Cech.
The stage looked set for an upset but Anelka had other ideas as he moved into the middle alongside Drogba.
Anelka hooked in the equaliser from a corner before settling Chelsea nerves by converting a centre from Ashley
