Tottenham, fifth in the Premier League and looking forward to the Champions League knockout stages, conceded two penalties and a red card in the first 14 minutes at Craven Cottage and were 4-0 down to Fulham by half-time.
Arsenal played just over half the match a man down after Sebastien Squillaci was sent off and they were being held by Huddersfield until Cesc Fabregas came off the bench to score an 86th-minute penalty.
Manchester City needed a goal from Edin Dzeko 10 minutes from time to earn a 1-1 draw at third division Notts County while Nenad Milijas missed a last-minute penalty for Wolverhampton Wanderers as his side lost 1-0 to Stoke City.
Nigeria striker Victor Obinna hit a hat-trick as West Ham United beat Nottingham Forest 3-2.
The fifth-round draw also took place on Sunday and produced a classic tie as Crawley, the first minor league club to reach the last 16 for 17 years, were handed a dream trip to Manchester United.
Orient were also rewarded for two upset wins with a home tie against Arsenal and holders Chelsea, should they win their home replay with Everton, will host Reading.
DEFLECTED SHOT
At Fulham, Tottenham's Alan Hutton felled Clint Dempsey to allow Danny Murphy to slot the opening spot-kick after 10 minutes and four minutes later Michael Dawson also pulled back Dempsey, earning a red card, giving Murphy the chance to net his second penalty.
Brede Hangeland tapped in the third and, after shambolic defending by Spurs, semi-finalists last season, Moussa Dembele lashed in a superb fourth.
Nicklas Bendtner's deflected shot after 22 minutes had Arsenal seemingly cruising but things changed when Squillaci was sent off three minutes before half-time for a professional foul.
Huddersfield made all the running after that and, after a series of near-misses and good saves by Manuel Almunia, they levelled with a header from a corner by Alan Lee.
Arsenal had needed a stoppage-time penalty to earn a 1-1 home draw with Leeds United in the last round and were handed a similar escape route when Bendtner was shoved and Fabregas, on for the last 20 minutes, dispatched the spot-kick.
Wenger was pleased with the "tough win" but was more concerned with a hamstring injury to Samir Nasri, who is now doubtful for the Champions League match against Barcelona on February 16.
"I took the gamble on Nasri - it backfired... he looks to have a serious hamstring injury," Wenger said. "Usually it is three weeks, I think he will miss the Barcelona game."
Manchester City, the richest club in football, were in danger of humiliation as Notts County, struggling in League One and with a squad made up predominantly of free transfers and trainees, matched their millionaire rivals in every department.
County deserved their 60th-minute lead via Neal Bishop's glancing header and were untroubled at the back until Micah Richards escaped down the right and crossed for 27 mill
