The 26-year-old has been intensive care since Wednesday after having a kidney removed following a heavy challenge from Forest Green Rovers striker Julian Alsop on Tuesday night.
Second half goals from Carl Heggs and Matthew Williams gave Tamworth their first ever Conference victory over Boro.
Heggs stabbed home from close range on 62 minutes after defender Peter Folkes had flicked on a Michael Tuohy corner.
Eleven minutes later, Williams fired home a stunning free-kick from 25 yards out to record his third goal in as many games for the Lambs since joining them on a month's loan from Notts County.
Williams could have scored almost immediately after Heggs' strike when he robbed Rob Quinn and raced through on goal but Alan Julian produced a great save.
Boro had shaded a drab first half and were unlucky not to take the lead just before the break when Louis Jefferson's header from Justin Gregory's cross came back off the bar.
He went close with an almost identical header ten minutes after the restart but Tamworth were hardly troubled after that and won the game comfortably in the final half hour.
The terrible news about Bevan had rocked the Lambs dressing room and manager Mark Cooper feels it has galvanised his relegation-threatened squad.
"It is terrible that something like this has happened to such a nice lad," he said. I am sure Scott was in everyone's minds today and the players were determined to do well for him."