Barcelona forward Luis Suarez scored the only goal at the Rostov Arena, sweeping in from close range on his 100th appearance for his nation.
La Celeste flattered to deceive against a side which was routed 5-0 by hosts Russia in the tournament opener, however, with the Saudis enjoying more possession than their opponents.
"The only thing that surprised me was the low level of my team in the first half, we were very far from our usual level, and you could say that's because Saudi Arabia made it hard for us," Tabarez told reporters.
"Some people might think that winning these matches is easy. People were talking about Russia, Egypt and Saudi Arabia without really knowing.
"In this World Cup we've seen so many surprises so far that we can't take things for granted. Remember in the past when we saw scoreboards of 10-1 in a World Cup? You don't see that very much anymore."
Uruguay currently sit second in Group A behind Russia, with both sides on six points and assured of a place in the last 16.