Anthony Martial scored the only goal of the contest in what was a close, if lacklustre, encounter at Turf Moor - in turn cutting City's lead at the top of the pile to single figures ahead of their game with Newcastle United this evening.
The result extended United's unbeaten league run to eight games and also saw them notch up their fifth clean sheet on the bounce in all competitions.
"Turf Moor's a tough place to come - they throw a lot of bodies forward, they're a big physical side and we needed to stand up to that. To a man we did that today," Jones told reporters after the game. "The character, the determination, the fight, the willingness to close down today was great.
"[Getting clean sheets] gives us a great platform to go and win the game and hopefully we can keep doing that. We're picking up points, we need to keep doing that, and who knows where we'll go to.
"For sure this puts pressure on [Man City]. We were 12 points clear in my first season at United... it's going to be difficult [to overtake them], but never say never."
Next up for Jose Mourinho's side is a trip to Yeovil Town in the fourth round of the FA Cup on Friday night.