Romelu Lukaku scored twice on the opening day after his £90m move from Everton while Nemanja Matic introduced himself to the Old Trafford crowd with a man of the match display.
But the Swans have won two of the last three meetings between the sides in South Wales, and Britton believes they can upset the odds again.
“I watched Manchester United last weekend and they were very impressive,” said Britton.
“They have spent a lot of money in the summer and it looks like their signings have had an impact straightaway.
“But we have proved many times that we can give anyone a game at home and that’s what we’ll look to do this weekend.”
Mourinho’s men ran riot to beat West Ham 4-0 after Swansea began the campaign with a hard-fought draw at Southampton.
“We have proved many times that we can give anyone a game at home”
Swansea captain Leon Britton
Paul Clement masterminded the great escape for the Welsh side by taking 19 points from a possible 27 on offer at the Liberty Stadium.
And Britton wants to lay the foundations for a successful season by continuing their stellar home form.
He said: “Since the manager came in at the beginning of January, our home form has been very good,” Britton added.
“It wasn’t there in the first half of last season, but we then produced in the second half of last year and that was a big reason why we stayed up.
“In every season where we have been successful since I have been at the club, our home form has been crucial.
“We have always picked up points away from home along the way, but home form has been the key.
“That is where you can put the foundations down for a good season and we want to do the same again this season.”