The pair saw defending champion Djokovic and Spain's Alcaraz suffer shock defeats at Flushing Meadows earlier this week.
Top seed Sinner had no such problems against Christopher O'Connell, beating the Australian 6-1 6-4 6-2 in less than two hours to reach the second week in New York.
Meanwhile, world number five Medvedev claimed an assured 6-3 6-4 6-3 win against Italy's Flavio Cobolli.
"Everything can happen in this sport. I try to stay on my side of the net and try to take it day by day," Italy's Sinner said when asked about Djokovic and Alcaraz's early exits.
Russian Medvedev added: "I said before the match that for me the upsets don't matter. The conditions here are tricky and the favourites have fewer margins."
The 23-year-old will face American home hope Tommy Paul, who fought back from a set down to beat Canadian qualifier Gabriel Diallo 6-7 (5-7) 6-3 6-1 7-6 (7-3).
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