The home side, who were missing Fernando Llorente through injury, dominated the first half, with Tom Carroll’s long-range shot just wide.
The loss of Gaston Ramirez through injury reduced Boro’s creativity with Adama Traore’s pace their main threat.
In the second half Victor Valdes, who was the busier goalkeeper, denied Gylfi Sigurdsson’s long-range shot.
Swansea put Boro under late pressure as Sigurdsson’s deflected free-kick brushed a post while Rudy Gestede headed a great stoppage-time chance wide for Boro.
The draw moves Swansea one point above Hull City in 17th, while Boro remain 19th, five points off safety but with a match in hand.