Substitute Gestede slotted home Boro's first league goal in 509 minutes of action to give the relegation-threatened side hope after Marouane Fellaini and Jesse Lingard had put the visitors in control with first-half goals.
But in the closing stages there were ugly scenes as Gestede and Bailly clashed, with television replays showing the forward moving his head towards United defender's neck, as if attempting to bite him.
The Benin international played down the incident, in which he grabbed Bailly around the neck as his opponent pushed his face into Gestede's shoulder, and insisted there was nothing more malicious from either player.
"We had a few words together, it was nothing major," Gestede said.
"He gave me a hug and a kiss, that's it. I didn't feel anything.
"Ben [Gibson, Boro defender] thought he bit me, so that's why he reacted like he did. But I told him now we were just talking, there was no problem."
Rows between players continued in the tunnel after the full-time whistle, but managers Jose Mourinho and Steve Agnew – standing in after this week's sacking of Aitor Karanka – denied having seen any incident.
Bailly did not reference the clash when posting on social media following the game, writing on Twitter: "We knew how to work seriously and suffer until the last minute. Three points well earned."