Spot-kick specialist White converted the all-important penalty seven minutes before the break, after Isaac Vassell was brought down by Eastbourne keeper Lewis Carey as he tried to go past him after cutting in from the right.
The visitors' No1 saw red and stand-in keeper Bouwe Bosma had no chance as White found the net to give City a priceless lead in the difficult conditions.
White's penalty in the week had given City a 1-0 FA Trophy win at Cirencester Town and of course his spot-kick won the promotion play-off final at the end of last season .
The home side though were unable to kill the game off in the second period, even after Craig Stone was also sent off in the final ten minutes for picking up a second yellow card.
And the home fans had to endure a few anxious moments, with a Jack Evans free-kick rattling the bar, as the visitors pressed for an equaliser, before they could celebrate the win.
For the game, Tully made two changes with Vassell and Cody Cooke recalled and Jake Ash and Craig Duff dropping to the bench.
With the strong wind and heavy pitch making good football difficult, there was little between the sides in the opening stages. For City, Jamie Richards headed over Ryan Brett's free kick and Cooke also found the net, but it was ruled out for offside, while at the other end Evans forced a good save out of Martin Rice with a free-kick and Darren Lok shot just wide.
Vassell was the greatest threat for City with his searing pace and it was that pace that helped set up what proved to be the game's only goal, as he burst clear down the right and was then fouled by keeper Carey ,who received his marching orders from referee Steven Plane in the turning point of the game.
White calmly dispatched the spot kick, leaving Truro, perhaps a trifle fortunately, ahead at the break in what had been until then had been an even game.
At the start of the second half, Matt Wright was denied by the legs of stand-in keeper Bosma who was not really tested a much as he should have been.
Eastbourne were far from finished and after Evans's free kick had shaken the bar they continued to cause problems for the home defence.
City needed a second, but it just would not come.
Even when Stone saw red for pulling back substitute Les Afful, after earlier being booked for dissent, City still could not relax, with substitute Cameron Gill, on his City debut after signing on loan from Exeter City, shooting over and Vassell hitting the post in a terrific goalmouth scramble, both unable to put the result beyond doubt.
In the end though City managed to see the game out and will now look forward to the double header against Bath City in confident mood.
Truro City: Rice; White, Pugh, Richards, Riley -Lowe; Knowles, Green, Cooke (Afful 64) , Brett (Ash 79), Wright (Gill 83), Vassell.
Subs (not used): Duff, Mitchell.
Yellow cards: Cooke, Knowles, Pugh
Eastbourne Borough: Carey; Stone, Beale, Evans (Haysman 77), Simpemba, Khinda-John, Hamilton, Collier, Romain (Pinney 60), Lok (McCallum 60), Bosma.
Subs (not used): Baptista, Worrall.
Red cards: Carey (38), Stone (82 (two yellows).
Yellow cards: Simpemba.
Goal: White (pen 38) 1-0.
Man of the match: Isaac Vassell - his pace terrorised the Eastbourne defence and won the crucial penalty which in the end decided the game.
Referee: Steven Plane (Worcester).
Read more: http://www.westbriton.co.uk/Truro-City-v-Eastbourne-Borough/story-28394330-detail/story.html#ixzz3uwgIqhG2