The Uruguayan full-back joined in the summer from Penarol but was involved in the Under-20 World Cup with his country before finalising his arrival in Manchester.
Philip Dean shocked United by handing the Evo-Stik League side an early lead with a ferocious shot from the edge of the area in only the second minute.
Although Joe Rothwell forced Grant Shenton to push over his shot, things were not looking good for the Reds when a lovely ball by Mark Drew picked out Tom Brookes, who despatched a great goal on the turn past Kieran O'Hara for a second goal.
Back came the visitors with Lawrence reducing the arrears thanks to a well-taken penalty after he was fouled inside the box. Welsh youngster Lawrence struck again after collecting a pass by Jack Barmby and United were unlucky not to go in ahead at the break with Varela and Barmby both going close to scoring.
After O'Hara was forced into a good save to thwart Brookes, Ramsbottom shuffled their pack with five substitutes and soon fell behind for the first time thanks to Lawrence's dipping drive over Shenton's outstretched arms.
Barmby got in on the act with a carbon-copy strike to give the Reds some breathing space with Varela's work done after 64 minutes as Donald Love took his place. As the midfield took complete control of proceedings, United slipped into exhibition mode and were unlucky not to score a fifth when James Weir's blistering long-range drive missed the target by inches.