The Argentina forward secured a narrow 2-1 victory at Craven Cottage, sweeping a wonderful 25-yard strike into the top corner ensuring an eighth-straight league win.
City leapfrogged Arsenal to the top of the table – with a game in hand on the Gunners – while they remain well on course for a potential treble, with a Champions League semi-final against Real Madrid and FA Cup final with Manchester United also on the horizon.
And the World Cup winner encouraged his team-mates to keep pushing in pursuit of their aspirations for the campaign.
"It was a very important win for us today," he told BBC Match of the Day. "We need to keep this going.
"We knew it would never be easy coming here because of the pitch, because of how good Fulham are as a team. We were very happy to withstand the pressure and hold out.
"[Top of the table] is where we wanted to be at the start of the season. Everyone's put in really good work, and now we've reached the top, we need to defend it.
"Every game is a final for us and everyone's going to pull together to make that final effort."
Kyle Walker also acknowledged City cannot afford to relent in their charge, but the England defender dismissed treble talk as he is instead focused on the upcoming showdowns with West Ham and Leeds United.
"[Fulham] was always going to be a tough game after the emotions of Wednesday [against Arsenal]," Walker told BBC Radio 5 Live.
"I think the form is so important. It has been good over the recent couple of months, but we have so much football to be played, and this form needs to continue.
"This is what you live for as a footballer – to be competing at the highest level. I'm not even concentrating on the treble. It's just the next game and getting ready to face West Ham. We need to get another three points at the Etihad and then move on to Leeds."
The campaign has been suspended due to the coronavirus pandemic, which has brought sport to a standstill around the world.
It remains unclear when, and if, the Premier League campaign will resume, but Walker is hoping the season can get back underway.
"Football has to take a back step for now because that is the least important thing on everyone's mind in the football world," the right-back said.
"The health of family members and other people's family members is the most important.
"But, I hope, and I am sure everybody else hopes, that football will come back and we will be able to finish off the season, but if it is not a possibility, then everyone does understand that people's health is more important than a game of sport."
Walker discussed how City – who were second to Liverpool in the Premier League when the season was stopped – were keeping up team spirit using group chats.
But the 29-year-old revealed manager Pep Guardiola, who donated €1million to the coronavirus fight in Spain, was not in the players' chat.
"Obviously Pep is not in the group chat! I dropped him a text message last night saying hope you are OK, and I hope the family is well and fair play for donating the amount of money that you have donated," Walker said.
"That says a lot about him as a person. Put football aside, that has come from his heart and it is something that should definitely be recognised."
City signed Haaland from Germany's Borussia Dortmund, while he previously played in Austria for Salzburg and his native Norway for Molde.
Although his link-up play has occasionally been criticised, Haaland's goalscoring talent has endured across every league.
He has 27 goals already in the Premier League, closing on the single-season record of 34, but Guardiola expects it would be a similar story in any division.
Asked if Haaland was particularly well suited to playing in England, Guardiola replied: "And I would say in Germany, in Spain, in Italy, in Norway, in Congo, everywhere.
"These types of players adapt everywhere; in Africa, South America, no problem."
Haaland's goals tally might have been even more impressive had Kevin De Bruyne performed on a slightly more consistent basis this season.
Guardiola is sticking by the midfielder, however, acknowledging his team since arriving at City has been "Kevin and 10 more".
"If Kevin is fit, he is ready," the manager said, also confirming Kyle Walker would be available to face Crystal Palace following an incident at a club that was captured on CCTV. The England defender appeared to pull his trousers down, with Cheshire Police looking into the matter.
City will hope to beat Palace on Saturday and again apply pressure to league leaders Arsenal, who responded last week by dramatically beating Bournemouth.
That result was huge for Arsenal, but Guardiola insisted it had no adverse impact on City.
"Of course we take a look. We have to do our job and take a look," Guardiola said. "You see it's 0-2 and after 2-2 and after it's six minutes extra time, they score after.
"So, it's a pity. It would be better with a draw.
"We have to admit we are used to this for Liverpool. How many times in the last few seasons they scored in the last minutes at Anfield? How many?
"It was one of the biggest talents for Liverpool to scored in the last minute. I've lived it in the last three, four years, when we fought against them and battled many, many times."
City said they were "disappointed" following Sunday newspaper claims that Walker hosted a party with a friend and two women at his home in Cheshire on Tuesday.
The club said Walker's behaviour "directly contravened" their efforts to support key public services during the pandemic, and said he would face a disciplinary investigation.
Walker's alleged breach of social distancing guidelines follows a separate episode involving Aston Villa midfielder Jack Grealish, who apologised after visiting a friend last weekend.
City said in a statement: "Manchester City FC are aware of a story in a tabloid newspaper regarding the private life of Kyle Walker in relation to a breach of the UK lockdown and social distancing rules.
"Footballers are global role models, and our staff and players have been working to support the incredible efforts of the NHS and other key workers in fighting the effects of the COVID-19 coronavirus, in any way we can. Kyle's actions in this matter have directly contravened these efforts.
"We are disappointed to hear the allegations, note Kyle’s swift statement and apology, and will be conducting an internal disciplinary procedure in the coming days."
Walker, 29, has made 21 Premier League appearances during the current campaign, his third season at City.
The England full-back won back-to-back titles during his first two seasons after joining the club from Tottenham.
He expressed regret for stepping out of line amid the allegations against him.
"I want to take this opportunity to issue a public apology for the choices I made last week," Walker told The Sun.
"I understand that my position as a professional footballer brings the responsibility of being a role model.
"As such, I want to apologise to my family, friends, football club, supporters and the public for letting them down."
Clubs in Saudi Arabia and MLS are reportedly keen on Kyle Walker, who has submitted a transfer request at Manchester City.
Fellow Red Devils attacker Jadon Sancho has also been omitted from the 25-man squad, having also been left out of Southgate's last party in November 2021.
Rashford has endured an underwhelming season at Old Trafford, registering a paltry four goals and two assists in 19 Premier League appearances as his team fights for a top-four finish.
The 24-year-old's frustrating campaign has led to suggestions he could leave Old Trafford, and he described himself as "upset" and "disappointed" after being heckled by his own fans following Tuesday's Champions League defeat to Atletico Madrid.
Meanwhile, 21-year-old defender Guehi could make his Three Lions debut in the Wembley friendlies against Switzerland and Ivory Coast, having impressed for Patrick Viera's Eagles this season.
Guehi has made 28 Premier League appearances since leaving Chelsea for Selhurst Park in July, with the Eagles keeping eight league clean sheets this term, a tally betted by just seven Premier League sides.
Along with Rashford and Sancho, Manchester City defender Kyle Walker is another big-name absence, with Southgate favouring Trent Alexander-Arnold and Reece James in his right-back position.
Alexander-Arnold and James have been two of the Premier League's three most creative defenders this season, having recorded 11 and six assists so far this term, respectively (Liverpool's Andrew Robertson has registered 10).
England host Switzerland on March 26 and Ivory Coast three days later, as the Euro 2020 runners-up ramp up their preparations for this year's World Cup in Qatar.
The Three Lions overcame their Euro 2016 tormentors when a turgid affair exploded into stoppage-time drama – Raheem Sterling winning and converting a 91st-minute penalty as Sverrir Ingason joined Walker in receiving his marching orders before Birkir Bjarnason missed a subsequent spot-kick for the hosts.
England's increasingly laboured attempts to secure victory before that point were thrown into flux when Walker, already booked, launched into a rash challenge on Arnor Ingvi Traustason with 20 minutes remaining.
A mainstay of England's run to the semi-finals of the 2018 World Cup, Walker did not feature for his country last season.
In that context and considering competition from the likes of Trent Alexander-Arnold, Kieran Trippier and the as-yet uncapped Aaron Wan-Bissaka, the 30-year-old Manchester City defender acknowledges his error could come with significant ramifications.
"To get sent off for your country is never a nice moment, especially after being out of the team for so long," Walker told Sky Sports.
"I have to take full blame. I'm an experienced player and I should know when I'm on a yellow card not to make a rash challenge like that.
"It's unacceptable from me.
"[Wanting to impress] possibly plays a big part in it but I don't want to blame that. I'm a seasoned professional and I've played a lot of games.
"So I should know how to manage a game if I'm on a yellow card. I put myself in a situation I shouldn't have been in. I think you have to say fair play to the lads for digging in and fighting and getting the win we desperately wanted."
Walker's City team-mate Phil Foden made his England debut with an assured showing in midfield, while Manchester United teenager Mason Greenwood came off the bench for a first senior cap – underlining the ever-evolving options at Southgate's disposal.
"It's brilliant for English football," Walker said. "Throughout the age groups they've gone and won things.
"Phil was involved in the [2017] World Cup win for the under-20. They're proving it at club level for big clubs.
"The world's their oyster. Obviously I know Phil - and Mason seems to be a very good player. He's sharp and can finish so I wish them all the best."
He added: "That's the sad thing about being a footballer. It does come to an end sometimes and I'm probably near the end on my international career
"I still feel very energetic, I feel like I've got a lot in the legs. Hopefully I will be back. It hurts."
