The Whites were only able to win 1-0 through Diego Llorente's opportunistic finish at a corner, but their dominance over Watford was far greater than the scoreline suggested, and Bielsa described the victory as "deserved" Leeds hit the crossbar through substitute Tyler Roberts ont he 88th minute.
"Of course I feel relieved," said the Argentinian coach. "Three points from a possible 18 (as Leeds had before kick-off) is something to be worried about, independent of what we deserved or not.
"To win today was a necessity."
Reflecting on his side's performance, he said: "There were a lot of positive things - we didn't make errors, we didn't allow counter-attacks, we didn't let their forwards stand out. In the first hour of play we created a lot of danger, in the last 30 minutes a little bit less.
"It was a deserved win and the difference could have been bigger.
"There were some players who shone.
"I thought (Jamie) Shackleton evolved and Diego Llorente came back well (after injury).(Kalvin) Philips had influence, Tyler Roberts came on and played well and the rest were all at a level from medium to high."
Bielsa revealed that Crysencio Summerville missed the game through illness and that whilst Jack Harrison was on the bench, he was only to be used in an emergency because of a pain in his oblique muscles.
Stuart Dallas came off injured during the second half, but there was no prognosis yet on the problem.
Bielsa's opposite number Xisco reflected: "We know exactly the intensity of them, they had more intensity than us, they shot more times than us, they more aggressive than us and they wanted to win the game more than us.
"We know Leeds are No 1 in the Premier League for intensity and aggression, all the numbers.
"We only shot one time on target. It's poor."