The result means that Ole Gunnar Solskjaer's side remain unbeaten on the road in the Premier League as both sides failed to close the gap on league leaders Liverpool.
United should have opened the scoring inside the first two minutes but Rashford's close-range header went over the bar.
The England striker made up for his sitter after 23 minutes when he was set up by Fernandes to break the deadlock.
The hosts grew into the game and equalised on the half-hour mark when Barnes sent a long range shot past David de Gea.
United came close to retaking the lead before half-time but Fernandes was unable to convert a header from Luke Shaw's free kick.
United came alive after the hour mark when Rashford was put through on goal and his shot produced a brilliant save from Kasper Schmeichel.
Anthony Martial thought he put the visitors back in front moments later but his strike was ruled out correctly for offside.
Substitute Edinson Cavani set up Fernandes to score late on but Vardy's shot was deflected in off Tuanzebe to earn a point for Leicester.
"Disappointed we didn't win today because we had loads of big chances and we could've finished it off but they're a dangerous side," Solskjaer said after the match. "One point is not the worst result but we're disappointed to not get the three points against a tough side."
"I think it's good to be disappointed," Ferandes told MUTV. "I think, today, I was really happy when I got into the dressing room because everyone was disappointed and saying the things that we have to do better. I think it's good as it's creating a mentality, a winning mentality.
Leicester boss Brendan Rodgers praised the way his team had approached the game and responded to the two United goals.
"I thought it was a good game -- two attacking teams and I thought we had lots of really good play," he said. "Our defending and counter pressing was very good you have to do that against teams like Man United they break so fast."
Rodgers added: "We go 2-1 behind and we've always had enthusiasm and confidence but these games you go behind against a team that's been playing so well it really showed the mentality of the players and we deserved a point at least out of the game."