Hamilton is on 66 poles, meaning another in Spielberg on Saturday would have set him up to equal the record at Silverstone, his home British Grand Prix, next week.
Hamilton laid down the mother of all markers in FP1, topping Vettel's best time on supersoft tyres by nearly half a second, on the hardest compound available this weekend, softs.
Hamilton won 12 months ago, emerging less scathed from a last-lap tangle with team-mate Nico Rosberg to end the German's previous grip over the event.
Bottas was 0.370s adrift of his Mercedes teammate in third place after suffering a spin at Turn 1, while Vettel was also within half a second of the fastest time.
Hamilton was leading the race in Baku but was forced to pit midway through on safety grounds after his headrest came loose. "The balance of the auto feels good and we'll fine tune the set-up some more to be ready for qualifying". "I lost three or four tenths in the last sector, so the auto felt definitively quick".
"This track punishes you if you make a mistake, and I tested those limits, but that's something I can learn from", the Finnish driver said.
Hamilton, victor in Austria past year, is 14 points behind Ferrari's championship leader Sebastian Vettel after eight races but still has a good chance of going into the August summer break in the lead. Not only did the Belgian outpace Fernando Alonso, but McLaren looked competitive with Honda's Spec-3 engine, which has been introduced ahead of the week.
Vettel was given a time penalty during that race but escaped further punishment from the FIA, motor sport's governing body, following a hearing on Monday.
Earlier on Friday Hamilton led the way in practice, clocking the fastest time in both sessions as he sets out to close the gap on title rival Sebastian Vettel.