The 52-year-old victim was attacked in Fowler Close on Saturday morning, and despite police officers administering CPR, he died at the scene.
A 64-year-old man has been arrested on suspicion of murder and remains in custody.
Police said they were called shortly before 11am to reports of two men fighting in the back garden of a block of two-storey flats. Neighbours said that they believed there had also been an argument the night before.
Detective Inspector Lee Holehouse, who is leading the investigation, said: “We are in the very early stages of our investigation but we can reassure the public that our initial findings are that this is a contained matter involving two individuals that are known to each other.
“I am mindful that there was a separate, unrelated, incident in Wellington last night and that these two incidents happening within 24 hours may, understandably, cause concern. Thorough investigations will be carried out for both and you will see an increased presence from police across the town for your reassurance.
“We are carrying out house to house enquiries in the area but we are asking anyone who witnessed the incident, or any activity leading up to it, that hasn’t yet spoken to us to get in touch.”
On Sunday Fowler Close remained coned off and there was police tape outside one property and part of the green in the street.
Saturday's tragedy came just hours after a man in his 20s was stabbed following a three-vehicle road crash in Holyhead Road. That incident, which police say is unrelated, happened at around 8pm on Friday night.
Four men were later arrested in Market Street in relation to the incident. The street was taped off for several hours and police sent a helicopter up to assist officers on the ground in dealing with the incident.
A 45-year-old man was arrested on suspicion of attempted murder, while the other three men, aged 20, 25 and 28-years-old, were arrested on suspicion of violent disorder.
Superintendent Mo Lansdale said: "Following two very serious but unrelated incidents in Wellington this weekend, I'd like to offer reassurance to the community that our investigating officers are dealing with the recent events with upmost severity and professionalism, to bring justice to those that have caused harm.
"I don't underestimate the impact that incidents such as these have on people and their loved ones, and I encourage anyone with any concerns to contact their local policing teams, whose contact details can be found on our website.
"You may see more police officers around Wellington and Telford over the weekend, they are there as part of our commitment to you, to be visible and accessible when you need them the most.
"I'd like to thank you for being understanding of the restrictions and road closures we have had to put in place, and for your continued support whilst our enquiries continue. We hope to remove all road closures as soon as possible and will update you when we can."