The Blues bounced back from Jay Rodriguez's goal after 13 seconds to beat Southampton 3-1 and move within four points of leaders Arsenal on a day when Liverpool, Manchester United and Tottenham all dropped points.
Mourinho played down Chelsea's title credentials after goals from Gary Cahill, John Terry and Demba Ba saw his side win their first of nine December games, merely suggesting it was important his team were in the mix.
"They must feel the responsibility to play for Chelsea," said Mourinho, whose side next play Sunderland, Stoke and Crystal Palace before the December 23 clash with Arsenal at the Emirates Stadium.
"Feeling that Chelsea cannot be in December - like it happened in a few seasons - completely out of the title race.
"I can't establish targets. I can't say 'now I want to win five consecutive matches, now I want to be top of the league then'.
"One of the things these boys have to learn is to cope with the responsibility of being there. We cannot be 15 points behind by December, we have to be there."
Mourinho's unbeaten Premier League home record was under threat after Michael Essien's error allowed Rodriguez in, but the Portuguese's run extended to 67 games courtesy of a second-half fightback.
"The goal was like a knife in our back; by surprise you are losing 1-0," Mourinho said.
"(But) in the first half we played very well. We played with balance, we played calm, no panic, no silly decisions, no mistakes, not going from 1-0 to 2-0, not risking a lot. That was a sign of maturity.
"We deserved the three points in a match which was a must-win for us. We knew that three title contenders lost points before us, so it was important for us to win this game."
Ba was Essien's replacement and proved a nuisance up front alongside Fernando Torres in a 4-4-2 formation as Chelsea chased the game, with Frank Lampard on in midfield for Oscar, who went off injured.
"We gave Southampton a different game, a game that they didn't expect," Mourinho said.
"I think it was difficult for them to cope with the situation."
Mauricio Pochettino admitted the early goal had an impact on Southampton's approach and rued the concession of two goals from set-pieces.
"It's clear that the game changed when we scored that goal so early on," Pochettino said.
"We conceded two goals from set-pieces. That cannot happen.
"That's a mistake that shouldn't happen in football. They were soft goals we shouldn't have conceded."
Artur Boruc had done well to deny Chelsea, but the goalkeeper departed after Cahill's equaliser and went to hospital for X-rays on his hand and wrist.
Medical tests subsequently revealed a fractured hand that is expected to rule the Poland international out of action for six weeks.