Weber rebounded from a crash that destroyed the front wing of his car on Friday to time a lap of 1 minute, 27.030 seconds - the fastest across the Formula One races three practice sessions.
Renault's Alonso trailed the Australian pilot by 0.142 seconds in second while Weber's Red Bull teammate David Coulthard was third in 1:27.193.
Nico Rosberg of Williams was fourth and Ferrari's Felipe Massa, the two-time defending champion in Turkey who was fastest on Friday, rounded out the top five.
Toyota's Italian driver Jarno Trulli, enjoying his racing this year, was sixth quickest ahead of Britains Lewis Hamilton in his McLaren with Germany's Timo Glock eighth in the other Toyota.
Honda's Jenson Button and Brazilian driver Nelson Piquet, in his Renault, completed the top 10 at the end of the hour long session on a track left greasy by early morning rain.
Kimi Raikkonen, who is coming off his second win of the season at Barcelona, was 11th in his Ferrari. The defending world champion leads Lewis Hamilton of McLaren by nine points in the overall standings but has struggled with his cars setup since a gearbox problem in Fridays first session.
The Turkish GP has been moved up from its usual date in the hot days of August, leaving teams grappling with wet weather instead of the usual sunshine. More rain is expected ahead of Saturday afternoons qualifying for Sunday's race.