Alonso was happy with the point against a spirited South Korean side who took the game to Uruguay before the South Americans found their rhythm and threatened with creative moves and precision passing in an end-to-end Group H match.
Veteran captain Diego Godin, playing in his fourth World Cup at 36, produced a strong performance at the back and came close to scoring when his first-half header hit the woodwork and Federico Valverde struck the post in final minutes.
"It as a tight game between two competitive nations. First 20 minutes was South Korea, we struggled, we couldn't take the ball from them but we could adjust," Alonso told reporters.
"Second half we were better, we attacked a bit more, tried to be more competitive. I liked a lot of things the players did today."
The teams are joined in Group H by Portugal and Ghana, who also met on Thursday, with the European side winning a 3-2 thriller.
Uruguay were composed in attack, with Godin and Jose Maria Gimenez feeding long lofted passes into the path of Facundo Pellistri and an industrious Darwin Nunez, who battled for a goal on his World Cup debut but struggled to get past South Korea's resolute defence.
Uruguay play Portugal on Monday and face Ghana in their final group match on December 2.
"I'm more than happy with the performance of my players and I'm sure this group stage will be determined at the last match," Alonso said.
"Obviously, the second game is key as is the third. Whether we qualify or not depends on us."
Real Madrid's Valverde, who was left out of Uruguay's final squad for the 2018 tournament, was voted man of the match on his World Cup debut, which he said was a dream come true.
"I really enjoyed this match, I was really anxious, it was a beautiful experience...were all very sad not to have won," he said.
"My team mates gave it their all to fulltime. If we keep working like this we'll go far."
He commended South Korea for their ability to disrupt Uruguay's moves.
"In midfield we always want to have control of the ball, South Korea pressured very well and sometimes we were marked, we had no passing opportunities and that really hindered our mobility," he said.
(Editing by Ed Osmond)
