Early in the second half Robinson got one back for the Blades before Chelsea defender Kurt Zouma deflected the ball into the net following a dangerous cross from Robinson.
Belief for Blades
Robinson told the BBC: "At half-time the gaffer just told us to believe a bit more. Coming to Stamford Bridge after just coming up, it was all new to us but we have to believe, that's what the gaffer said. We have proved we can play against the best.
"I missed a chance at a crucial time to make it 1-1 but we stick together. Chelsea have players with a huge amount of ability but we dug in, pressed them when we had to do and to a man we worked hard for each other."
Robinson off the mark
Robinson, who signed for the Blades from Preston North End in the summer, got his first goal for the club to get United back into the game two minutes into the second half with a composed finish after a great cross from Enda Stevens.
Following that United pressed Chelsea higher up the pitch and made sure they didn't have many chances. Abraham nearly completed his hat trick but Dean Henderson pulled off a fantastic save.
Robinson turned provider in the final minutes when his cross came off Zouma and flew past Kepa sending the travelling United fans wild.