Liverpool's season has imploded in recent weeks, with Juergen Klopp's side winning just two of their previous 11 games in the league, but there were green shoots of recovery in a dominant performance against their struggling opponents.
The result left Liverpool in sixth position in the table on 43 points, one behind their next opponents Chelsea. Sheffield United remain bottom on 11 points.
With Liverpool's Brazilian goalkeeper Alisson on compassionate leave following the death of his father this week, Adrian took over between the posts but it was Blades keeper Aaron Ramsdale who shone in the first half with some excellent saves.
There was little he could do to stop Jones early in the second half, however, as he blasted home Firmino's backheel three minutes into the second half.
Firmino thought he had scored Liverpool's 7000th top-flight goal when his 64th-minute shot took a heavy deflection off Bryan before flying into the net to make it 2-0, but the effort was subsequently given as an own goal by the defender.
Striker Oliver McBurnie had a couple of decent chances for the home side but despite displaying plenty of energy, they could not manage to mount much of a fightback as Liverpool held on for their first league win since beating West Ham United 3-1 on the last day of January.
"I'm happy that we get the points and we can go again. Today was a good example of putting it all together, I am happy I scored and I'm happy for the rest of the team," Jones said after dedicating his goal to the memory of Alisson's father.