The 19-year-old Shaqiri, who was born in Kosovo and moved to Switzerland as a child with his parents, reaffirmed his potential as he revived Switzerland's hopes of qualifying for Euro 2012 via the play-offs.
Switzerland, handed a lifeline on Friday when second-placed Montenegro lost 2-1 to Wales, fell behind when Ivan Ivanov scored at the far post following a corner in the ninth minute.
The Bulgarians spent the rest of the first half time-wasting and paid for it when Shaqiri levelled in the four minutes of first-half stoppage-time added for some farcical play-acting.
The FC Basel player, described as a street footballer by his club coach Thorsten Fink, played a clever one-two with Blerim Dzemaili before collecting the ball in a crowded area and sweeping it home with superb control.
Shaqiri, one of three members of the Swiss team which finished second at this year's European Under-21 Championships in the starting line up, put the Swiss ahead just after the hour with a curling, left-foot shot.
Having been booked for a shirtless celebration, he completed his hat-trick in the last minute with a long-range shot and even Switzerland's usually dour coach Ottmar Hitzfeld celebrated joyfully on the touchline.
The visitors had Zhivko Milanov sent off for a second bookable offence in the 65th minute.
"It was a very good performance by Shaqiri, he could have had three goals before he had even scored his first," said Hitzfeld. "Tactically, he was very good."
Switzerland are third in Group G with eight points from six games, three points behind Montenegro who did not play on Tuesday. The Swiss host Montenegro next month.
