The tie finished goalless in normal time, with both sides hitting the post, alongside missing presentable chances.
The goal was enough to see Enzo Maresca’s side through to the last eight for the first time since they won the tournament in 2021.
The defeat stretches Bournemouth’s winless run to six games in all competitions but they can now concentrate on securing their Premier League survival, with Andoni Iraola’s side currently eight points clear of the drop zone.
“It was a good win and a good game,” Maresca said
“I don’t think winning tonight shows we are Premier League level or if we’d lost that we aren’t Premier League level. It is just a game. Tonight we showed our level and we are happy.
“QPR on Saturday [in the league] is the most important thing in this moment.”
Iraola, whose side had 27 shots on goal, said: “It is disappointing and it was a bad night for us because at the end it was the worst thing that could happen.
“We had a lot of shots and chances to finish the play and in the end if you play 120 minutes and don’t score you deserve to go out.
“I think it’s an opportunity missed.”
Bournemouth spurn chances as Leicester dream continues
On Saturday, Bournemouth went toe-to-toe with Manchester City but just lacked the finishing touch to earn a point or more against the Premier League champions.
This was a very similar situation.
Both sides attacked with real intent and pace in the early exchanges but the hosts created the better openings with Luis Sinisterra a livewire down the right.
His cutback saw January signing Enes Unal have a shot blocked on the line, before another pass from the Colombia winger was put over the bar by Alex Scott.
Leicester grew into the half and come closest of all when Marc Albrighton’s low drive cannoned back off the post, before Wanya Marcal’s curling effort was tipped over.
Bournemouth defender Marcos Senesi hit the same post in the early exchanges of the second period, with a left-footed cross drifting over everyone before connecting with the woodwork.
The second half was a more cagey affair but Leicester’s Yunus Akgun had a superb chance to win the tie when through on goal with three minutes remaining but he skied over the bar.
The winner came from the one piece of brilliance in the game and continues what has thus far been a memorable season for Leicester, who are six points clear at the top of the Championship.
With three months remaining, it could yet be an unforgettable one for Maresca and his side.
