The Warwickshire champions trailed twice in the first half, but with fly-half Gary Challis pulling the strings in midfield, Street took a firm grip to leave their visitors pointless in the second half.
James Hewitt kicked two penalties to give Bs a 6-0 lead in 14 minutes before Street showed their attacking flair to score tries in quick succession through centre Alex Croft and winger Dan Phillips, the second converted by Challis.
The visitors were back in front by the half hour, going in for a converted try, but Challis secured a 15-13 lead at the break with a penalty and then added another soon after the restart.
Challis kicked his third penalty on 59 minutes as Street turned the screw, and after converting when winger Craig Ross gathered his own chip to cross for his side's third try, the No.10 sealed victory and a personal 16-point haul with a final penalty six minutes from time.