Vogel, former assistant to Thorsten Fink, had no experience as a senior coach when he took over in October after his fellow German left for Bundesliga side Hamburg SV.
He was thrown in at the deep end as Basel lost 2-0 at home to Benfica in the Champions League less than a week later.
But they went on to take seven points from their final three games, stunning last year's runners-up United 2-1 on Wednesday to qualify for the knockout stage.
"The sum of all the key criteria such as sporting success, social competence, professionalism and Heiko's development allowed the FCB Basel club management to make a unanimous and happy decision," said the club on its website.
Vogel, 36, a former minor league player, worked with Bayern Munich's youth divisions for nine years, then as Fink's assistant at lower division German club Ingolstadt.
He also won his first six league games with Basel, taking them top with a seven-point lead at the halfway point.
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