The close tie looked destined for a replay until Michael Fernandes gave the Vanarama National League South side the lead four minutes from time.
An unforgettable afternoon was capped deep into stoppage-time when Francis Amartey made sure as the Boro bridged a 51-place gap.
"I'm delighted for everyone involved in the club, it's been a long time coming," beaming boss Day told BBC Radio Surrey.
"It was a big day and the support was amazing. The players put on a fantastic performance against a very strong Sutton side so I'm absolutely thrilled for everyone involved.
"We defended well and I thought we had some good chances and I think we've missed the best chance of the game when Deering got through.
"We were waiting for a moment and we set it up well and it was one of those days where things went our way."
The shock result means Farnborough are in the Second Round for the first time since the club reformed in 2007 - something not lost on Day.
"When you've been in football a long time like I have you have to enjoy these moments," Day - who has been in charge at the Hampshire club for over a decade - said.
"These days don't come around very often. They [Sutton] have had an incredible decade here under [Paul Doswell] and Matt Gray.
"It was a really tough game and we're thrilled. I think we're eight unbeaten now.
"We'll enjoy it but then we're back to the obligatory on Tuesday night. I'm not going to get involved in the draw - what will be will be."