The Foxes welcomed Luciano Spalletti's side to the King Power Stadium in the opening group game, and Ayoze Perez gave them an early lead with a close-range finish.
Barnes, who put in the cross for Perez's opener, then doubled his side's lead soon after the hour mark, but a Victor Osimhen brace in the final 20 minutes saw the visitors draw level and forced Brendan Rodgers's side to settle for a point.
Speaking to reporters after the game, the winger said: "They are a very good side and we knew it was going to be a tough game. But after playing some really good football, we will be disappointed. You don't want to be chasing points. You want to get them early. There's a little bit of a bitter taste not to come away with the win."
In the next round of group fixtures, the Foxes travel to Poland to take on Legia Warsaw, who defeated Spartak Moskow on Thursday to immediately take top spot in Group C.
Udinese 
Torino 
Napoli 
Venezia 
Leicester 
Manchester City 
Everton 
Nottingham Forest 
Tottenham 
Wolverhampton 
Celtic 
ST Johnstone 
Motherwell 
Rangers 
Juventus 
Fiorentina 
West Ham 
Liverpool 
Sampdoria 
Pisa 
AC Milan 
Roma 
Sporting CP 
Benfica 
Ipswich 
Chelsea 
Manchester Utd 
Newcastle 
Brentford 
Arsenal 
Inter 
Atalanta 
Nacional 
FC Porto 
Valencia 
Real Madrid 
Aston Villa 
Bournemouth 
Crystal Palace 
Almeria 
Sevilla 
Bordeaux 
Brighton 
SC Braga