Electing to bat first after winning the toss, the visitors had slumped to 36 for four before Amit (76 not out) and Mukesh (73) resurrected the innings with their defiant stand.
Goa's medium pacer Gaurish Gawas earned his maiden first class wicket when he had Himachal skipper Aakash Chopra (2) trapped in front of the stumps.
Varun Sharma (15), Paras Dogra (6) and Nikhil Gangta (9) too departed in quick succession as the visitors' decision to bat first on a seaming track backfired badly.
Amit and Mukesh then combined well and revived the team's sagging fortunes.
Amit has so far struck 13 fours in his unbeaten 197-ball knock while Mukesh's 179-ball stay at the crease was studded with eight fours and a six.
At stumps, Rishi Dhawan was giving company to Amit on 18.
For Goa, Gawas and left-arm spinner Shadab Jakati took two wickets each.
Brief Scores:
Himachal 1st innnings: 210 for five in 90 overs (Amit Kumar 76 not out, Mukesh Sharma 73; Shadab Jakati 2/56).
