Osorio, ranked No 53, beat Japanese Naomi Osaka (6-4, 6-4) in the previous round of the Indian Wells tournament and thanks to her relentless game and defense, she also posed problems for Tauson who employs a similar brand of tennis to Osaka.
At 4-5, Tauson was down two set points in her own serve, but she remained cool in the decisive moments which has been one of her trademarks in the new season to take her own serve and eventually close out the set by taking the tiebreaker 7-3.
Losing the first set left Osorio visibly deflated and Tauson raced to a 5-2 lead in the second set before the Columbian regained her composure and brought the scoreline back to 5-5.
But then Tauson, who is one of the most improved players on the WTA Tour in this season, upped the pace again and forced Osorio into making a host of unforced errors which was enough for the Dane to book her ticket for the next round of the tournament.
Tauson, ranked No 21, will play the winner of the match between French contender Varvara Gracheva and Russian Mirra Andreeva, the No 9 seed, next.