Ian Blackwell contributed 191, John Francis 123 and Matthew Wood 95 as Durham UCCE struggled.
However, on a good pitch, Durham UCCE fared well, making a mammoth 304 for the first wicket, and were 327 for 1 at close on the second day.
Somerset chose to bat out the rest of the day for a draw, finishing on 211 for 2 declared, with John Francis scoring 112.
(Cricinfo scorecard) Match drawn Leicestershire batted first at Grace Road, and Darren Maddy made the most of his chance against the students.
On the third day, Durham UCCE went on to score 255 thanks to 73 from David Balcombe, avoiding the follow-on and saving the game.
Durham made a good go at winning the match, though, making 258 for 7 declared to set the students 308 in 74 overs.