At stumps on the second day, Aakash Chopra was batting on 24 and Varun Sharma was unbeaten on 34 as the visitors negotiated the day's final phase without losing a wicket.
For the hosts, Pallavkumar top-scored with a 90-run knock, while Amit Sinha made 53.
Tarjinder Singh and J Syed Mohammad contributed 40 and 47 respectively.
Rishi Dhawan was the best Himachal bowler, returning figures of four for 60.
The match is being played at Nehru Stadium.
Brief scores:
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Assam 1st innings: 354 all out in 117.5 overs (P Das 90; Rishi Dhawan 4/60)
Himachal Pradesh 1st innings: 62 for no loss 30 overs.