Tottenham boss Mourinho engaged in a pre-match war of words with Fulham manager Scott Parker over the Premier League's decision to reschedule the London derby at short notice.
Harry Kane had put the hosts ahead in the first half at the Tottenham Hotspur Stadium, but Cavaleiro struck in the closing stages to leave Mourinho frustrated.
Tottenham have won just two of their last eight league matches and sit in sixth place.
A win against lowly Fulham would have taken them into third place above Manchester City, who made it four successive league victories when Phil Foden earned a 1-0 win over Brighton earlier on Wednesday.
City are unbeaten in 14 matches in all competitions since losing at title rivals Tottenham in November.
Whether Spurs -- six points behind leaders Manchester United -- will be regarded as title contenders for much longer must be in doubt after this sloppy display.
Third-bottom Fulham have drawn five successive league games and are just two points from safety with two games in hand.
Cavaleiro's leveller was a suitably dramatic end to the feud between Mourinho and former Tottenham midfielder Parker, triggered when Aston Villa's coronavirus outbreak meant Spurs' scheduled trip to Villa Park on Wednesday was postponed.
Instead, Tottenham's game against Fulham -- originally called off in December because of Covid cases in the Cottagers squad -- was moved on Monday to help ease a potential fixture pile-up for Mourinho's team.
Parker said it was "scandalous" and "blatantly wrong" for his team to be forced to play at such short notice.
Mourinho hit back, saying he would only feel sorry for Fulham if they played a weakened team.
"If they come with their best team, I think they should apologise to all of us," Mourinho said.
Mourinho would have noted with interested that Fulham named a strong team with nine of the 11 that played their last Premier League game on Boxing Day.
Fulham looked capable of troubling Tottenham right from the start and Andre-Frank Zambo Anguissa's fierce strike forced a good save from Hugo Lloris.
Tanguy Ndombele's cross picked out Son Heung-min for a volley that Fulham keeper Alphonse Areola saved at point-blank range.
Areola made an even better stop to push Son's curler over the bar, but Tottenham struck first in the 25th minute.
Harry Winks' sweeping pass found Sergio Reguilon on the left wing and he whipped an inch perfect cross into the six-yard box, where Kane dived to power a spectacular header past Areola for his 18th goal of the season.
Not even the prolific Kane is perfect and the Tottenham star squandered a good chance to double the lead moments later when he headed over from Serge Aurier's cross.
Fulham remained a threat and Kenny Tete volleyed just wide of the far post.
Pierre-Emile Hojbjerg was guilty of a bad miss when he shot straight at Areola from Kane's pass.
Winks' powerful drive from the edge of the area was well saved by Areola.
But Son held his head in his hands after the South Korea forward surged clear, only to see his low strike hit the far post and rebound to Areola.
It proved a crucial moment as Fulham snatched an equaliser in the 73rd minute.
Ademola Lookman sent over a superb cross from the left side of the Tottenham area and Cavaleiro got between Reguilon and Eric Dier to plant a brilliant header past Lloris.
Tottenham were rocking and Lloris had to make a good save to stop Ruben Loftus-Cheek.
Reguilon had a late goal correctly ruled out for offside to ensure Parker finished with the greater feeling of satisfaction.