Romelu Lukaku's first-half double put United in control before the second half briefly took a dramatic turn when Burnley goalkeeper Joe Hart saved Paul Pogba's penalty kick and substitute Marcus Rashford was sent off for violent conduct only 11 minutes after coming on.
"Even with 10 men for 15 minutes the team was the team that had the chances to score," Mourinho said. "We are all very happy that the boys can go for their international week with the feeling (of winning)."
Tottenham, which set alarm bells ringing at Old Trafford by winning 3-0 there six days ago, was unable to carry that momentum to Watford, which came from behind to secure a 2-1 victory at Vicarage Road.
Watford has won all four of its Premier League games and joins Liverpool and Chelsea with a maximum total of 12 points at the top of the table.
Arsenal, meanwhile, joined United on six points as it secured a first away victory under Unai Emery with an entertaining 3-2 win at newly promoted Cardiff.
After storming out of his news conference following United's defeat to Tottenham, demanding "respect, respect" from the media, Mourinho was much calmer ahead of his side's trip to Burnley, and the approach may have brought rewards on the field.
Mourinho will have been especially pleased by United's improved defensive performance as it claimed a first clean sheet of the league season.
Both the Portuguese and his squad will enter the international break with confidence that United's worst Premier League start for 26 years is firmly behind them.
SMALL DRAMAS
The only dramas for United this time concerned Pogba's penalty miss and Rashford's red card, but Mourinho wasn't critical of either.
After Rashford was fouled in the box, Pogba — who has already scored two penalties this season — had his spot kick saved by Hart.
"I never blame a player for missing penalties," Mourinho told BBC Sport. "I blame the ones who refuse to go up there. Paul's scored some good ones for us."
The frustration of the penalty miss appeared to affect Rashford more than Pogba and within a minute of the miss the England striker had been sent off.
Burnley defender Phil Bardsley tackled Rashford before kicking him in the aftermath of the challenge. Rashford attempted to take matters into his own hands, pushing his forehead into Bardsley's face, and was sent off.
"I would say it was naive," Mourinho said. "It was a kid and a very experienced man. Bardsley's been in the game for 20 years and Marcus is a naive boy."
SPURS STUNNED
With Tottenham having taken the headlines last round with a statement victory over Manchester United, some seemed to forget that both teams held perfect records as Spurs made the short trip to face Watford on Sunday.
By the end, there was only one perfect record to talk about — Watford's.
The win for Javi Gracia's team was made all the more impressive by the fact Spurs took the lead shortly after halftime, as Abdoulaye Doucoure found his own net as he attempted to clear Lucas Moura's cross.
The usually reliable Spurs defense would have been confident of seeing out victory from there, but was undone in the simplest fashion.
Troy Deeney and Craig Cathcart both headed in set pieces in the space of seven minutes to turn the match around and keep Watford level with Liverpool and Chelsea at the top of the table.
GUNNERS DUO STRIKE
It's still early days in Emery's reign as Arsenal manager, but Sunday's win at Cardiff may prove a pivotal moment for the Spaniard.
For the first time this season he opted to include both Pierre-Emerick Aubameyang and Alexandre Lacazette in his starting lineup and was rewarded with a first away victory in the Premier League.
While there are still major defensive concerns for Arsenal, the play of the strike duo suggests Emery has a combination that will cause problems for any defense.
Shkodran Mustafi gave Arsenal an early lead before Victor Camarasa leveled with Cardiff's first goal of the season in first-half stoppage time.
Aubameyang struck after the break as Lacazette flicked the ball into his path, allowing the Gabon forward to find the bottom corner from outside the box.
However, Arsenal was pegged back once more as Danny Ward headed the home side level with 20 minutes remaining.
Not to be denied, the inspired Lacazette hit an unstoppable effort in at the near post with nine minutes remaining to finally see off Cardiff, which is still looking for a first win of the season.