The black BMW 640i was later found in south Bradford and police are appealing for witnesses who may have seen the gang flee on foot near the Spen Valley Greenway and Low Moor railway station.
The burglary happened at an address in Park Drive, Farnley, Leeds, at around 2.40pm on Wednesday.
The vehicle, which was displaying the registration LD65 JOV, failed to stop for officers as it drove towards Drighlington and was found a short time later in Cleckheaton Road in Bradford, where it had been locked and secured.
Police enquiries led to the arrest of two boys aged 15 and 17 from Bradford. They have since been released under investigation as enquiries continue.
Detective Sergeant Jonathan Druett, said: “We believe there were four or five occupants in this vehicle when it drove off from Farnley and would like to hear from anyone who saw it, or who may have footage of it on their dash cam as it headed towards Bradford.
“This vehicle was seen driving at speed along Bradford Road and then on to Toftshaw Lane towards the M606.
“On exiting the vehicle we believe the suspects travelled on foot through Victoria Park towards the Greenway and then crossed the lines near to Low Moor railway station, so we would also appeal for anyone who may have seen people acting suspiciously in this area at around 3.20pm.”
Anyone who can assist this investigation is asked to contact police on 101, quoting crime reference 13180353992, or use the live chat facility on the West Yorkshire Police website.
Information can also be given anonymously to the independent charity Crimestoppers on 0800 555111.