Morrison equalised for the U's with 17 minutes to go and then ran more than half the length of the pitch to celebrate in front of the travelling Blades fans.
The Blades dominated for long periods and Matt Heath cleared Stephen Hunt's effort from underneath his own crossbar early on, but the visitors went ahead in the 17th minute through John Cofie, who collected possession more than 25 yards out before firing in a looping shot over goalkeeper Mark Cousins.
Cofie inexplicably blasted over from five yards and Cousins was forced to save Blades striker Nick Blackman's low shot early in the second half.
But Colchester equalised when Morrison converted from close-range after Magnus Okuonghae had helped on Anthony Wordsworth's corner in the penalty area.
Colchester substitute Andy Bond struck the inside of the far post late on and Cousins tipped Blades defender Tony McMahon's header over as the spoils were shared.