The pair were at the tail of the field after the initial 90-minute session at the Hungaroring, but despite never previously rounding a corner in a F1 car, 19-year-old Alguersuari – whose full name is pronounced 'Heimi Al-gay-shuari', started cautiously but drove professionally.
There was one anxious moment, however, when his Ferrari-powered car stopped in the middle of the pitlane and had to be pushed by his jogging mechanics back to the garage.
Two seconds further up the time sheets was pacesetter and 2008 winner Heikki Kovalainen, and in a further sign McLaren is set for a competitive weekend, the third best time of the morning was set by his teammate Lewis Hamilton.
The improving Mercedes powered car was also quick on the similar Monte Carlo layout earlier this year, and two weeks ago in Germany a big upgrade moved Hamilton's pace to within half a second of the pace.
"It is like starting a new season," Hamilton said in Hungary on Thursday. "I can't imagine how good the feeling would be for the entire team to get on that podium again."
The Brawns didn't feature near the top of the order in morning practice, but Red Bull's Sebastian Vettel seemed to be faring even worse, explaining over the radio that the RB5 was overheating the tires under the Budapest sun.