Diego Costa scored midway through the second half as the Blues won for only the fourth time in 13 Premier League matches on Saturday, and for the first time in four games.
Mourinho said that the ending to the match was nervier than it needed to be given his side's dominance, but praised the attitude of his side in the victory.
"Me and the fans did not deserve to have our heart in our hands in the last four minutes," he told BBC Sport. "It is the pressure of the bad results, that is normal, but they coped well.
"The only thing that we did not do was score more from the chances we had. We should have scored at least two and breathe easier but it is difficult for us to score goals and to keep a clean sheet against a team who plays defensively.
"Diego Costa missed two big chances in the first half but he scored and he could breathe easy. If the result is fake, it is fake because we should have scored more goals."
The result lifted the defending champions to 14th in the Premier League table.