Sainz set a best time of one minute 13.013 seconds, 0.030 quicker than Norris, who took to the track after a morning drama involving a fire in the team's paddock hospitality unit and evacuation of staff.
Ferrari's Charles Leclerc was third, 0.007 slower than Norris, and the Monegasque made contact with the McLaren towards the end of the session in what looked like anger at being delayed.
"He just drove into me. I think I have got damage," said Norris over the radio. Both faced a meeting with stewards after the session.
Red Bull's triple world champion Max Verstappen was fourth fastest and just 0.074 off Sainz's pace.
Mercedes pair George Russell and Lewis Hamilton, who had been quickest on Friday, were fifth and sixth with Red Bull's Sergio Perez seventh.
Stewards were also investigating an incident between Hamilton and Aston Martin's Lance Stroll after they made contact.
"This guy thinks he is alone on track," Stroll fumed about Hamilton.