Nat Borchers had put RSL up 1-0 in the 68th minute when he took a Morales free kick and sent it off the right post past Matt Reis for his second goal this season.
New England's Kenny Mansally made it 1-1 in the 82nd minute when he poked the ball into the goal after RSL goalkeeper Nick Rimando stopped a shot by Sainey Nyassi.
The goal ended Rimando's shutout streak at 511 minutes.
Saborio then answered with his 10th goal.
With the victory, Real Salt Lake (14-4-9) jumped atop of Major League Soccer's overall standings with 51 points, one up on Los Angeles, which plays at Chivas USA Sunday.
New England fell to 7-15-5 with the loss.