It was a fortunate equalizer for the USMNT, who had squandered their best opportunities during the first half before falling behind.
Brian Rodriguez's first international goal had Uruguay – without the likes of Luis Suarez, Edinson Cavani and Diego Godin – on track, but Morris levelled for Gregg Berhalter's men.
A beautiful team move led to Uruguay's first chance in the sixth minute, but Jonathan Rodriguez was just unable to get a shot off with only USA goalkeeper Brad Guzan to beat.
After Cristian Roldan headed straight at Fernando Muslera for USA, the hosts created a great chance midway through the first half.
Morris beat his marker down the left before crossing to Tyler Boyd, who put his shot into the ground and over the crossbar from close range.
A Tim Ream block denied Uruguay an opener moments later before Muslera pushed away a 25-yard effort from Sergino Dest six minutes prior to the break.
USA should have been awarded a penalty on the stroke of half-time for a handball by Jose Gimenez, but they went into the interval locked at 0-0.
Brian Rodriguez broke the deadlock five minutes into the second half thanks to his first international goal.
A counter-attack saw Federico Valverde play a pass to Brian Rodriguez, who beat his marker before firing a power left-footed effort past Guzan from inside the area.
USA were fortunate to equalise with 11 minutes remaining, a clearance by Uruguay coming off Nick Lima and falling to Morris to chest in at the back post.
What does it mean? USA avoid three-game losing streak
Runners-up at the CONCACAF Gold Cup, USA were well beaten by Mexico last week and the draw with Uruguay at least meant they avoided being handed three straight losses for the first time since 2015.
Goalscoring woes go on for USA
USA must find a way to score goals and quickly. It took a huge piece of luck to get their equaliser against Uruguay and they have struggled to find the back of the net since thrashing Guyana and Trinidad and Tobago to begin the Gold Cup. Since those wins, USA have scored six goals in six games, netting more than once in a match just once in that run.
Uruguay show off depth
Without their two star forwards in Suarez and Cavani, Uruguay showed off some depth in the attacking third. Brian Rodriguez, 19, opened his international account and Jonathan Rodriguez was also dangerous.
USA begin their CONCACAF Nations League campaign with matches against Cuba and Canada next month, while Uruguay will face Peru in two friendlies.