However, Mumm went over in the dying moments to snatch victory for the Chiefs in a thrilling finale.
The home side started the brighter, and Andy Goode successfully kicked an early penalty to put the Warriors in front.
However, their lead lasted just a matter of minutes as Arscott went over under the posts before Steenson converted.
David Lemi thought that he had responded with a try for Worcester just moments later, but he was penalised for a knock-on.
Worcester did manage to reduce the gap shortly after, when Goode's penalty cut the difference to a point.
Chiefs added their second try of the match in the 21st minute with sustained fast handling yet again. Haydn Thomas made the half break before releasing Steenson for the try, who then converted well.
Goode had a chance to get the Warriors back in game on the half-hour mark, but his effort dropped short of the posts as Exeter took a 14-6 lead into the break.
Chiefs began the second half on the front foot, and Steenson found the middle of the posts with a massive kick to make it 17-6 to the visitors.
The Warriors reduced the gap to six points with a try from Kvesic after the hosts successfully won the lineout.
The home side then went in front when Drauniniu was played in out wide for the try, before Goode added the extras.
However, with four minutes left to play, Chiefs' pressure on the Worcester try line finally paid off when Mumm went over from short range, before Steenson added the conversion to secure the away side a 24-18 triumph.