After a brilliant victory against Armenian Levon Aronian in the previous round, it was a topsy-turvy battle against Caruana for Anand that was finally drawn.
Hikaru Nakamura of United States, meanwhile, regained sole lead, defeating Russian Sergey Karjakin. In the process, Nakamura became the first player from America and 10th overall to breach the 2800 rating barrier in live world rankings.
The other game between Aronian and Vladimir Kramnik of Russia ended in a draw.
With two rounds under the Classical format and five under rapid still remaining, Nakamura sits pretty in the tournament with five points, while Anand is in close pursuit a full point behind.
Kramnik is in sole third spot on three points, coming from three draws, while Karjakin, Aronian and Caruana are not so far behind the Russian with two points each.
For the purpose of determining rankings, the rules in the tournament are unique as each victory in Classical format earns two points compared to one in rapid, which will be held on the last day of the event.
Caruana surprised Anand early in the opening by allowing the Neo Grunfeld vide a relatively less-played move order. Soon it was clear that the Italian meant business as he grabbed Anand's rook for a minor piece by seriously compromising his development.
Anand won a pawn back to increase his compensation but Caruana defended valiantly, pushing Black back step by step after the trade of queens.
Anand had his task cut out and he stayed put with some wily manoeuvres even as the Italian seemed to have seized the control. The mistake from Caruana came on 24th move that met with a super blow from Anand and the Indian was back into the game with a bang.
However, he was in serious time pressure by this time and Caruana escaped with an easy draw instead of suffering for a long time.
Nakamura played a gamble and it paid off big time.
Playing white, Nakamura employed a dangerous but deeply analysed variation in the English opening. Karjakin's attempts to find out the correct moves in a very wild and complex position proved futile.
When the game ended in just 27 moves, Nakamura still had an hour left on his clock.
Kramnik faced the Catalan opening from Aronian and it was a topical variation on board in the middle game. Kramnik had to defend unpleasant position for a long time after he lost a pawn but the technicalities were far from easy for Aronian.
The draw was a just result after 61 moves.
Results Round 3: Fabiano Caruana (Ita, 2) drew with V Anand (Ind, 4); Levon Aronian (Arm, 2) drew with Vladimir Kramnik (Rus, 3); Hikaru Nakamura (Usa, 5) beat Sergey Karjakin (Rus, 2).
Image: Viswanathan Anand
Photograph: FIDE World Chess Championship