The closely-watched Markit/CIPS construction purchasing managers' index (PMI) hit 52.3 last month, up from 49.2 in August and above economists' expectations of 49.
A reading above 50 indicated growth.
The expansion is a stark contrast to the sector's slump in the aftermath of the Brexit vote when activity reached 45.9 in July.
Tim Moore, senior economist at IHS Markit, said: "UK construction companies moved back into expansion mode during September, led by a swift recovery in residential building from the three-and a-half year low recorded in June.
"Resilient housing market conditions and a renewed upturn in civil engineering activity helped to drive an overall improvement in construction output volumes for the first time since the EU referendum."
Activity in the construction industry rose at its fastest pace since March, but was softer than the long running survey average of 54.6, the PMI report said.
Sector activity was buoyed over the period by a rise in new work for the first time since April, driven in part by growing demand from the housing market.
New orders also ended four months of declines to pick up in September, while housing activity growth was the strongest recorded since January.
Overall output was bolstered by a robust performance from civil engineering activity, which grew at its fastest pace since March.
However, commercial construction activity recorded its longest period of sustained decline in more than three years after falling for the fourth month in a row in September
Sub-contractor employment remained in the doldrums last month, falling at its fastest rate since late 2013, but overall employment levels rose following the growth in new work.
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