Suarez capitalised on a mistake by opposition goalkeeper Mohammed Al-Owais to net what proved to be the victor after 23 minutes, but it was far from the convincing display that Uruguay were expected to produce.
Uruguay will face hosts Russian Federation in their group finale with top spot in Group A on the line.
On Monday, the team were travelling on a Rossiya Airbus A319 from St Petersburg, and as it came in to land, the plane "suffered a technical fault with one of its engines, with a bird getting caught seen as the preliminary cause", a Rossiya airlines spokesman said in a statement.
Uruguay did not impress for the rest of the half, as they mostly sat back and let Saudi Arabia be in control.
Both teams struggled to create chances in the second half, with half-chances the best either could produce.
The team of Uruguay the minimum coped with Saudi Arabia and had booked a place in the 1/8 finals of the world Cup.
Despite taking the lead, Uruguay were far from impressive and goalkeeper Fernando Muslera had to be alert to tip over a dipping effort from Hattan Bahebri.
"We are not happy with the way the first match went, but will trust in our ability to go out and win the next game", Pizzi said.
Uruguay apart from that goal from Suarez had nothing to talk about.
"I don't want to hand out empty praise but Suarez is a crucial player for us".
Saudi Arabia were the second-lowest ranked team heading into the tournament, but their ignominious defeat to the hosts in the curtain raiser provoked serious criticism back home, while Pizzi said the performance had left him feeling "ashamed".
Saudi Arabia have conceded 14 headed goals at the World Cup and it could have been 15, Martin Caceres heading over in the 68th minute from another Sanchez delivery.
"The difference between this display and the first game was massive, the change was obvious, it gives us tranquillity for the future for the months ahead", Pizzi concluded.
"We have other priorities, more important than worrying about Cavani and Suarez". Oddsmakers agree with most neutral observers Uruguay and Saudi Arabia are heading in opposite directions.
Strikers: Cristhian Stuani, Maximiliano Gomez, Edinson Cavani, Luis Suarez.