Russian Time driver Bird lost his advantage off the startline as fellow front-row man Leimer (Racing Engineering) leapt into the lead using the medium-compound tyre, which gave him a better launch than Bird's hard rubber.
They ran divergent strategies as a result, with Leimer pitting from the lead on lap 12, despite suffering from a vibration at the rear of his car in the opening stint, switching onto four new hard tyres.
On the harder tyres, Bird stayed out until lap 20 to switch to four fresh mediums. However, his right-rear wheel proved tricky to attach and he was slow to rejoin, and his 36.6-second lead over Leimer was turned into a 4.1s deficit.
"What happened in the pitstop guys?" asked Bird rhetorically on the radio later. "That was a shame."
Despite cutting Leimer's lead to half a second with three laps remaining, Bird's softer tyres then seemed to wilt in the closing laps, and he could not get close enough to make a move. He finished 0.8s behind at the finish.
Bird also reported he had "too much understeer in the first and second Lesmos. He was just unstoppable compared to us. It was a fun race though."
Leimer now leads the championship with 153 points, eight ahead of Bird, while former points leader Stefano Coletti is now 18 points adrift.
"I want to focus race by race, and then we'll see what happens at the end of the season," said Leimer.
Coletti was running fifth when he received a drivethrough penalty for speeding in the pits, and would later retire with a handful of laps remaining.
Bird's team-mate Tom Dillmann finished third, over 5s in arrears, ahead of Stephane Richelmi and Julian Leal, who all ran long first stints.
James Calado suffered a terrible start but salvaged sixth position, chased home by fellow Brit Adrian Quaife-Hobbs.
Caterham duo Alexander Rossi and Sergio Canamasas, along with Vittorio Ghirelli, rounded out the point scorers.
It was a disastrous race for Carlin. Jolyon Palmer was in the mix for the podium until his left-rear wheel fell off at Lesmo 1 after his pitstop, while title-contending team-mate Felipe Nasr suffered a hydraulic leak that forced him out of a points-paying position.
Results - 30 laps:
Pos Driver Team Time/Gap
1. Fabio Leimer Racing Engineering 47m48.311s
2. Sam Bird Russian Time +0.806s
3. Tom Dillmann Russian Time +6.137s
4. Stephane Richelmi DAMS +9.902s
5. Julian Leal Racing Engineering +15.400s
6. James Calado ART +18.748s
7. Adrian Quaife-Hobbs Hilmer +24.007s
8. Alexander Rossi Caterham +30.276s
9. Sergio Canamasas Caterham +40.272s
10. Vittorio Ghirelli Lazarus +44.906s
11. Daniel de Jong MP +46.540s
12. Johnny Cecotto Jr Arden +47.382s
13. Jon Lancaster Hilmer +47.635s
14. Rio Haryanto Addax +48.003s
15. Sergio Campana Trident +50.194s
16. Rene Binder Lazarus +1m02.186s
17. Daniel Abt ART +1 lap
18. Dani Clos MP +1 lap