Best had earlier avoided a memorable moment of controversy when his attempted pass to Rovers goalkeeper Neil Sullivan from an uncontested drop ball struck the crossbar.
That all came during an entertaining Championship encounter in which the in-form Freddy Eastwood saw his header hit a post while Michael McIndoe also struck the woodwork.
Dean Shiels missed a great chance for a Rovers side who competed well but posed less of a threat going forward and subsequently saw their four-match unbeaten run come to an end, with the Sky Blues stretching their winning sequence to three games.
Coventry came close to breaking the deadlock when Eastwood sprayed the ball out to the left and raced into the box to head Patrick Van Aanholt's cross against a post.
Best left referee Rob Shoebridge less than impressed when he got far too much purchase on his attempted pass back to Sullivan, with the ball bouncing off the top of the crossbar.
A crowd of 19,221 were then on the edge of their seats in the final five minutes of the half as an inviting ball whipped across the face of goal by James Chambers failed to attract any connection before Best wasted two glorious chances at the other end.
Firstly, the Sky Blues striker fired against a post from the penalty spot having been brought down by Sullivan while chasing a pass from Eastwood.
But he then blew arguably an even easier chance, putting the ball over the crossbar from around four yards.
The home side started the second half in the same way they finished the first and it was the Sullivan-Best show yet again as the Rovers goalkeeper superbly tipped a well struck volley to safety.
Doncaster striker Shiels then fired over from close range in not too dissimilar fashion to Best at the end of the first half.
Chances were coming at either end, although it was Coventry who were applying more of the pressure.
Rovers survived the onslaught but McIndoe, playing against his former club, then struck the outside of a post.
And with the game appearing destined for a goalless draw, Coventry were handed their second penalty of the afternoon when Chambers brought down substitute Gary Madine in the area.
Substitute Morrison stepped up to fire home the resulting penalty with Sullivan less than ready.
And although a manic close followed, the Sky Blues held on to record their first Boxing Day triumph since at 2002 win at Highfield Road.