Belgium's Clijsters, playing in just her second match since retiring at the US Open in 2012, took it to the British number one before eventually falling 6-3 7-5.
Konta earned the first two break points of the match before sweeping a forehand winner off Clijsters's serve to end the first set, with the second tighter as the pair held their serves until the Briton claimed the win from an unforced error.
Konta will next face Tatjana Maria who earlier dashed Heather Watson's bid for back-to-back titles in Mexico.
Watson won her fourth WTA title and first for four years at the Mexican Open in Acapulco on Saturday, beating Canadian teenager Leylah Fernandez in an exciting final.
The victory lifted Watson to 49th in the rankings, her highest mark since the summer of 2015, but Germany's world number 91 Maria came out on top on Tuesday, winning 7-6 (4) 3-6 7-5 in a hard-fought contest which lasted two hours 39 minutes.
Former world number one Venus Williams also bowed out in the opening round, the 39-year-old beaten in three sets by Slovakia's Anna Karolina Schmiedlova.
Meanwhile, second seed Kristina Mladenovic recovered from a nightmare start and a break down in the decider to beat Chloe Paquet in an all-French first-round tie at the Open 6eme Sens in Lyon.
Mladenovic needed two hours and 12 minutes to overcome wild card Paquet 1-6 6-2 6-4 and faces Anna-Lena Friedsam in the next round. The German won 6-2 6-2 over Russian qualifier Anastasiya Komardina.
"It was definitely not an easy match to face a countrywoman and also a friend, even though I don't know as much about how she plays," Mladenovic said.
"It was very far from being perfect today, but I'm just pleased with how I stayed calm and composed and how I fought through to find solutions to get the win today."
Elsewhere in Lyon, sixth seed Jil Teichmann defeated wild card Clara Burel 6-4 6-3, but fellow seeds Daria Kasatkina and Viktoria Kuzmova needed three sets to get the better of Pauline Parmentier and Lesley Pattinama Kerkhove respectively.
