However the two other staffers to whom they spoke laughed at them and refused to escalate their issue.
Mr Luetchford, originally from South Africa, told the Evening Standard: "We were shopping and picked up some washing powder. My partner held my hand and gave me a kiss and then suddenly we heard this lady shouting, saying we could not do that here, that we should be ashamed of ourselves and we should get out. She was quite rude and dismissive to us. We told her she had no right to talk to us like that, and she responded 'you people have no right'."
"I come from South Africa and I have never had that happen to me before, even though people think that in Africa, it is more backward. I have a problem myself with public displays of affection, but we were not doing that, it was just a peck on the cheek," said Mr Luetchford, a manager at Soho's Ku Bar who has been in a relationship with Mr Pievani for 18 months.
"I found the dishwashing powder we were looking for and my boyfriend held my hand, told me he loved me and kissed me on the cheek. I should not be called disgusting or be shamed because of that."
He has since made a formal complaint to the supermarket giant, requesting that those responsible "receive more than a slap on the wrist".
Mr Luetchford took to social media to draw attention to the incident.
He wrote: "Just got some some serious #homophobic #abuse by some @Tesco employees in Brixton because my boyfriend kissed me on the cheek. Disgusted!"
A spokesman for Tesco said that the incident is being investigated but declined to indicate how long that would take.
She said: "Everyone is welcome at Tesco so we take any allegations of discrimination extremely seriously.
"Our Brixton Store Manager is looking into the incident and our customer service team, who are in contact with Mr Luetchford, will update him as soon as they can."
A spokesman for gay rights charity Stonewall said: "We have been working with Tesco since 2011 to help them support their LGBT employees, and to ensure they create an inclusive environment for staff and customers.
"We're sure Tesco will take this incident seriously and conduct a thorough investigation. Unfortunately, discriminatory behaviour is an everyday experience for lesbian, gay, bi and trans people. There is still more to do and our job is to help organisations and institutions improve."