Saha scored an unbeaten half-century to help Indians overcome trouble waters on Day 3.
The hosts, who had overhauled Indians' 247/9 (declared) by 39 runs at the end of Day 2, stretched their lead to 59 runs before pressing for a victory and putting the visitors in again.
All-rounder Cameron Green, whose magnificent hundred played a key role in helping Australia A recover from a difficult situation and then inch past the Indians, stayed unbeaten on a well made 125.
In the second innings, regular breakthroughs left the Indians vulnerable and raised the Aussie hopes of sneaking through to victory till wicketkeeper batsman Wriddhiman Saha came to the rescue.
Saha showed ample resolve through his unbeaten 100-ball 54 and also strengthened his claim of playing the first Test by taking Indians to safety.
While not playing the kind of knocks they would've liked, Hanuma Vihari (28) and first-innings centurion Ajinkya Rahane (28) too eat into enough time for the visitors to feel confident declaring at 189/9, leaving Australia A with the task of getting 131 runs for a win in 15 overs.
Mark Steketee (5/37) ended with an impressive five-fer while Michael Neser (2/41) and Green (2/12) chipped with a couple of wickets each.
Umesh Yadav (1/14) gave the Indians an initial breakthrough in the second innings. After Joe Burns' departure for a duck, Will Pucovski (23*) and Marcus Harris (25*) stitched a stand. Pucovski then got hit on his head by a bouncer and walked off the field retired hurt with the team physio. Both teams shook hands at the end of the 15th over.
Brief Scores:
Indians 247/9 decl in 93 overs (Ajinkya Rahane 117*, Cheteshwar Pujara 54; James Pattinson 3/58, Travis Head 2/24) & 189/9 decl in 61 overs (Wriddhiman Saha 54*, Ajinkya Rahane 28) versus Australia A 306/9 decl in 95 overs (Cameron Green 125*, Tim Paine 44) & 52/1 in 15 overs (Marcus Harris 25*, Will Pucovski 23*, Umesh Yadav 1/14, Mohammad Siraj 0/19) ends in draw.