The Hammers were pushed all the way by League One title-chasers Bristol City at Ashton Gate on Sunday before prevailing through a late goal from substitute Diafra Sakho.
On a weekend that saw a host of the Barclays Premier League's big guns fall to shock defeats, Reid admitted he and his teammates were delighted to have avoided their names being added to the list.
"In the first half we were under pressure, which was always going to happen when you come away to these places," said the No2, who was involved in a game-long battle with Bristol City's giant striker Matt Smith and centre-back Aden Flint, among others.
"They gave a good performance but fortunately for us we kept a clean sheet and Sakho came on and got the goal.
"Obviously we wanted to finish off the game in one go, get a goal and get the win and we did that. Even though we didn't play our best football, we got there in the end and the most important thing is we're in the next round."
"The manager was speaking beforehand about the amount of upsets and we didn't want to be one of them. We did all right and made the most of the opportunities that we created and are now in the hat for the last 16."
West Ham are one of just six top-flight sides who are guaranteed to be in the fifth round, while the likes of Chelsea, Manchester City, Southampton, Tottenham Hotspur and the Hammers' third-round victims Everton are all out.
The draw for the last-16 will be made live on BBC1's The One Show on Monday evening at 7pm, with West Ham taking ball No15.