India, set a victory target of 299 runs in 83 overs, finished with 177-4 before bad light forced the match to be abandoned as a draw after tea on the fifth day at the Chinnaswamy stadium.
Tendulkar was caught in the covers off debutant spinner Cameron White just 15 short of surpassing retired West Indian Brian Lara's tally of 11,953 runs as Test cricket's leading run-getter.
Even as he neared the record, the 35-year-old vehemently protested to the umpires about allowing play to continue in fading light and walked away in disgust after becoming White's first Test victim.
Umpires Rudi Koertzen of South Africa and Asad Rauf of Pakistan suspended play six overs after Tendulkar's dismissal, only to return 40 minutes later when the light improved marginally.
Tendulkar ended the game with a career tally of 11,939 runs and is expected to overtake Lara when the second of four Tests opens in Mohali on Friday.
Tendulkar, who walked in to bat with India struggling at 24-2 before lunch, put on 53 for the third wicket with Gautam Gambhir and 61 for the fourth with Venkatsai Laxman.
Some 15,000 home fans cheered every run by the 151-Test veteran, who defied the Australians for three hours to take India to safety.
Laxman (42 not out) and Sourav Ganguly (26 not out) saw out the remaining session which was marked by frequent interruptions for poor light before the match was called off with 10 overs remaining.
Australia, who resumed at the overnight score of 193-5, declared their second innings at 228-6 half-an-hour after the start to give their bowlers a minimum of 83 overs to dismiss India.
Shane Watson had made 41 when he was bowled by seamer Ishant Sharma, who claimed three of the six Australian wickets to fall.
Wicket-keeper Brad Haddin hit an unbeaten 35, while White was on 18 when skipper Ricky Ponting declared.
India lost in-form opener Virender Sehwag and Rahul Dravid as they slipped to 24-2 by the ninth over.
Sehwag was caught by Matthew Hayden at first slip off Stuart Clark for six, one run after being missed by Haddin off the same bowler.
Ponting then dived to his right at mid-wicket to remove Dravid for five with a spectacular catch as the batsman played an on-drive to an overpitched ball from Brett Lee.
Gambhir, who made 29, failed to take advantage of surviving two close appeals off a single delivery in the morning session.
As umpire Rauf turned down Clark's leg-before shout, Haddin threw down the stumps and the appeal was referred to the TV umpire.
Replays showed Gambhir had just made his ground as the bails went off.
But the Delhi opener was bowled by left-arm seamer Mitchell Johnson in the post-lunch session.