Mason Greenwood's first-half equaliser was just a consolation for United, who suffered only their second defeat to Leicester in 23 years.
Four-time runners-up Leicester have never won the FA Cup, having last reached the final in 1969.
Iheanacho now has nine goals in his last nine games – including five in his last two outings – and the Nigeria international wants to maintain his momentum.
"I just need to focus and keep working hard every day," he told LCFC TV. "Hopefully, they will keep coming.
"As a striker, that's what you need, so hopefully I'll keep going and hopefully we'll get to the next round.
"Hopefully the best is yet to come. I just need to keep working hard every day in training and hopefully everything will fall into place."
Iheanacho capitalised on two defensive errors from United, first seizing on Fred's poor back pass to open the scoring and then escaping Scott McTominay to head in and seal the win.
Tielemans made it 2-1 after the break following Greenwood's first-half leveller and the Foxes outplayed the visitors.
"It's history, and it feels good," Iheanacho said. "We're happy and I know the fans are happy as well.
"We prepared really well this week in training – how we're going to press and trap them. That's what we did, and it worked really well for us. We got the first goal, so it worked really well for us."
United's most realistic chance of silverware this season is now the Europa League, with Manchester City 14 points clear of them in the Premier League.
It was their first defeat in 30 domestic away games but after Fred's first-half mistake boss Ole Gunnar Solskjaer refused to blame the midfielder.
"We want them to build from the back," he said. "I think we've been really solid, we've been really good and we've built a foundation to play from with what we've done.
"Of course, sometimes it goes wrong, but that's acceptable. Everyone can make a mistake. We win together and we lose together: as a team."
Solskjaer also hopes the international break will help re-energise his squad.
He added: "They've been at Carrington every single day, working together and training together every single day. So a change of scenery will do some of them good.
"They'll come back for Easter Sunday (at home to Brighton) and be ready to go again."