Resuming on 55 for four and chasing a victory target of 216, India appeared dead and buried in the last hour before lunch but an 81-run ninth wicket stand between Laxman and Ishant Sharma set up the opening win of a two-Test series.
India had lost a flurry of wickets to slip to 124-8 but Sharma contributed 31 valuable runs in his partnership with Laxman, who was batting with a runner due to a bad back, before the hosts scrambled over the finish line in a frantic climax.
"One of the best, one of the most exciting Test matches I have played and I have played some 150 now," Australia captain Ricky Ponting said in the post-match presentation ceremony.
"The Ishant-Laxman partnership was outstanding, it made the difference," he added.
When play began, India needed 161 runs to win while Australia required six wickets in an absorbing Test but the ever-reliable Laxman kept his head as wickets tumbled, managing the tail to perfection in denying Ponting a first win in India as skipper.
Tuesday's innings adds to Laxman's reputation as a thorn in Australia's side with all but one of his five 150-plus scores coming against them, including a 281 in Kolkata nine years ago that led a following-on India to a stunning win.
"He is a remarkable batsman. If he is at the crease, whatever field you set, he can rotate strike, hit boundaries and keep the scoreboard ticking. He is really an amazing batsman... he was very special," India captain Mahendra Singh Dhoni said.
Going into the final day, local hopes rested on overnight batsman Sachin Tendulkar (38) but the prolific and talismanic batsman threw his wicket away with a poor shot to depart just when his team needed him most.
Earlier, Nathan Hauritz had made Australia's initial breakthrough by ending overnight batsman Zaheer Khan's 34-ball vigil.
Paceman Doug Bollinger then took over, removing Tendulkar and Harbhajan Singh, the latter wicket coming after Dhoni had ran himself out to compound the home side's crisis.
Laxman and Sharma, however, did not panic and played deliveries on merit as the hosts eventually completed what is the fourth-highest successful chase on Indian soil.
Dhoni echoed Ponting's sentiments in rating the match as one of the best he had played in and said the India-Australia rivalry was always capable of reaching such heights.
"Heartbeat goes up, a typical thing. I think that's what make Test cricket interesting," Dhoni added.
"When the match started, no one expected a result within four-and-half days but bowlers from both sides forced one." Khan, who was named man of the match for his eight-wicket haul, felt India had played marginally the better in clinching a conTest that had numerous momentum swings throughout the course of the five days.
"It was a great game of cricket. Australia fought really hard and we did just enough to be on the winning side," said the pacer, the laTest member of the 250-Test wicket club.
