Berbatov, who left London for United in August, outpaced the defence after 36 minutes, latching on to a lofted pass from Michael Carrick and firing past goalkeeper Ben Alnwick before celebrating the goal in front of the Spurs fans.
That put the home side ahead just after a deflected long range shot from Paul Scholes had cancelled out Roman Pavlyuchenko's fifth minute opener for Tottenham.
Championship Swansea City had earlier scored the day's biggest upset when they knocked out holders Portsmouth with a deserved 2-0 win at Fratton Park.
Portsmouth, who beat Cardiff City in the Cup final in May, had no answers against another Welsh side, Swansea winning with first half goals from Nathan Dyer, on loan from Portsmouth's arch-rivals Southampton, and a penalty from Jason Scotland.
"It is not nice going out, full stop," Adams said.
"It doesn't matter whether you're the holders. When you are out-fought and outplayed for 45 minutes then you are always likely to be dumped out of the Cup."
Along with Sunday's Liverpool v Everton clash, the United-Spurs match was the highlight of the fourth round with Redknapp, whose Portsmouth side beat United in the FA Cup at Old Trafford last season, looking for his fourth career FA Cup victory over United.
Pavlyuchenko's superb opening goal -- a glancing header from a Tom Huddlestone cross -- gave Spurs a surprise early lead.
