Sela had eliminated 24th seed Kyle Edmund of England, the world no. 24, in the previous round, but he was beaten 6-7, 4-6 by Baghdatis of Cyprus, who had earlier beaten another Jewish player - Diego Schwartzman, currently ranked a career high 17.
In the match between Sela and Baghdatis, both players looked nervous in the first set with each of them losing their service four times before the Cypriot prevailed in a tie-break.
Sela and Baghdatis failed to hold their serve twice in the second set, before the Baghdatis got the crucial break of serve to win the set 6-4 and the match in one hour 49 minutes.
This was only the third time in his career that Sela, currently ranked 97 in the world, had reached the third round of a Masters tournament and he ended a close fought game against Baghdatis - his best friend on the professional circuit - looking clearly disappointed at a lost opportunity against a player ranked five places below him.
Sela, 33 next month, seemed in danger of slipping out of the world's top 100, before this week in California, where he has shown the kind of inconsistent form that has characterised his career.
The Israeli had already packed his bags to leave America after being hammered 0-6 3-6 by America's Evan King, ranked just 200, in the second round of the qualifying tournament. But Sela sneaked into the main draw as a lucky loser after Uzbekistan's Denis Istomin withdrew at the last moment with a shoulder injury.
Sela celebrated by dispatching Germany's Peter Gojowczyk, ranked 50, in the first round 6-4, 6-4 before taking on Edmund.
He has earned nearly $50,000 prize money in Indian Wells and should rise to around 88 in the world standings, ensuring his automatic qualification for the French Open and Wimbledon.