Bassett were 4-2 up at half-time courtesy of Tom Titcombe (2), Misurtin Aras and Craig Smith, while Wroughton netted via a penalty and a fortunate deflection.
However, that did not tell the whole tale as Wroughton were reduced to 10 men early on and played most of the match a man down.
After the break Bassett increased their advantage through Luke Carter and Faulkner completed the win from the spot late on.
Boss Dave Terry said: “Russell has a bit of a nightmare sometimes and we wanted him to keep a clean sheet against Wroughton, but we gave a sloppy penalty which didn’t help him.
“So in the last 10 minutes to make it up to him, we asked if he wanted to take the penalty and he did.
“It was a very good penalty and he could be our regular taker now.”
Elsewhere in Division One, Malmesbury Reserves triumphed 5-3 against Chippenham Park Reserves and Intel were defeated 2-1 at Marshfield.
Meanwhile, Purton Reserves secured a 2-0 win over visiting Corsham Reserves in their first Wiltshire Premier game in a month.
The match was level at 0-0 at half-time, but Purton took the lead within moments of the restart, when Reece Gilliver latched onto a through-ball and tucked his shot away.
The hosts might have extended their advantage but Jamie Sheppard saw his drive rattle the woodwork.
In the closing minutes Purton finally wrapped up the win, substitute Terry Moore charging down a clearance, beating two men and setting up Tom McCulloch who tapped in.
“The first half was a bit scrappy and you could see both sides hadn’t played for a while,” said Purton boss Albert Dearing.
“We had a much better second half, we showed a good bit of quality and had two or three chances to finish it off before we actually did.
“I don’t think our keeper had to pull off a save in the second half so we weren’t under much pressure.”
Elsewhere in the division, Wilts Calne Town kept their stranglehold on top spot courtesy of a 4-1 win over Vale of Pewsey.
Chippenham Park’s pursuit of Purton in second suffered a blow as they were held to a 2-2 draw at home to Bradford Reserves.