Goals by Zlatan Ibrahimovic, Henrikh Mkhitaryan and Marcus Rashford eased United to a vital three points after drawing six of their previous nine Premier League games.
They are unbeaten in their last 21 league games but Mourinho's side, who moved above Arsenal into fifth place, are still four points behind Manchester City with a game in hand.
Liverpool, in third place, are six ahead of United having played two games more.
"We want to fight in the Premier League until it is mathematically impossible. The Premier League we cannot win, but Europa League we can," Mourinho said.
"We had lots of players that were not here today and the most important thing after the three points was to have no more injuries. The result was good. We resisted the results of yesterday when Manchester City and Liverpool won."
United, who are in the quarter-finals of the Europa League, were without first-choice goalkeeper David de Gea but his deputy Sergio Romero had an easy afternoon in the Sunderland sunshine.
With a Champions League spot available for the winners of the Europa League, that is arguably their more likely route back into the elite competition and they face a quarter-final, first leg at home to Anderlecht on Thursday.
They then host Premier league leaders Chelsea next weekend.
Read the full article on eurosport.co.uk: Mourinho - We will fight for top four, De Gea should be fit for Chelsea