In contrast to the mass postponements in English football due to a surge in positive Covid-19 cases, all four games in the top-flight in Scotland went ahead on Saturday.
United were hit by an outbreak earlier this week and needed to field a youthful side with four teenagers among the starting line-up.
However, they pushed Rangers all the way to protect Giovanni Van Bronckhorst's perfect record in the league since taking charge last month.
United's victory in their previous meeting in August is the only league match Rangers have lost in their last 57 league games.
And they nearly frustrated the Gers once more but for Scott McMann's handball from a corner.
Rangers captain Tavernier slammed home the resulting spot-kick, but the hosts still had to ride their luck for all three points when Nicky Clark's header came off the bar in stoppage time.