The Scot beat Moldova’s Radu Albot, who was trying to claim his 100th tour-level win, 6-4 6-3 in one hour and 42 minutes.
Murray had just 18 unforced errors to 35 for Albot, and never faced a break point on his serve while firing six aces.
Having won in straight sets for the first time since October, Murray tweeted: “I’d forgotten what that felt like. Bosh.”
Draper has lost just eight games through his first two matches of his Indian Wells debut.
Murray said: “He obviously plays really well. He’s not actually competed that much on the tour but, when he has done, he’s always performed really well. I think he’s going to be a top, top player for a long time, providing he stays healthy. It will be a good match.”
Linette, who reached the semi-finals of the Australian Open, is the highest-ranked player Raducanu has beaten since her run to the title in New York and she played sharp, aggressive tennis after falling 1-4 behind in the opening set.
The 20-year-old continues to be dogged by physical problems and arrived in California struggling with wrist pain and the after effects of a bout of tonsillitis.
Raducanu sought treatment for her right wrist during the first set, and said afterwards: “It’s manageable. I’m still able to win matches at this level. I’m feeling better than I was the other day. I would say I wasn’t at 100 per cent, but better.
“I’m enjoying competing again. I think this feeling of competition definitely beats being sick or injured or out. So I’m really just cherishing every time I get to step out on the match court.”
Top-ranked Iga Swiatek routed Claire Liu in 65 minutes to win her opening match on Saturday.
Swiatek, the defending champion who took over the number one ranking nearly a year ago, won 6-0 6-1.
Carlos Alcaraz, the second-ranked men’s player, defeated qualifier Thanasi Kokkinakis 6-3 6-3, Taylor Fritz, the defending men’s champion, rallied past Ben Shelton 4-6 6-4 6-3, while Jannik Sinner defeated Richard Gasquet 6-3 7-6 (2).
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