Alonso first scored with a thumping downward header in the 19th minute and then converted a penalty in injury time.
"We knew this was the most important match and we played very well," Alonso said. "Scoring the early goal really made things easier, so I must say we're happy."
The win was Spain's first ever over France in a competitive match in seven attempts. Spain will next face Portugal on Wednesday in Donetsk in the semifinals as it bids to win a third straight major title.
"When you lose a game there's always something missing, I'm not sure if it's possession or technical ability," France coach Laurent Blanc said. "The lads gave everything and it's very hard to play against Spain. We blocked their left flank, which is their strong point."
The opening goal was the result of excellent one-touch football — the hallmark of Spain's game. Andres Iniesta prodded the ball through for Jordi Alba, who whipped a perfect cross from the left to Alonso. The Real Madrid midfielder timed his run so well that no France defender picked him up.
"He knows how to sacrifice himself, help the team and where to play," Spain coach Vicente del Bosque said. "He is one of the attacking strengths of ours. He knows how to create and finish."
Spain wrapped things up after France right back Anthony Reveillere barged into substitute forward Pedro Rodriguez in injury time. Alonso coolly sent goalkeeper Hugo Lloris the wrong way from the spot, looking right but clipping the ball firmly to the left.
France's only shot on target came in the 32nd minute when Yohan Cabaye's free kick was tipped over by Spain goalkeeper Iker Casillas.
"They were there for the taking," Blanc said. "We knew we would get some chances, and the only thing I regret is that they scored off their first good chance. If we had gone in level at 0-0 at halftime, then I think we would have had more room in the second half."
The match was sometimes dull, but the Spanish did not really need to change gears after Alonso's opening goal.
"These types of games, in this type of competition, are always very tense, and you need to have the know-how to win them," Del Bosque said. "We were very tired at the end. We need to recover well because the 48 extra hours (Portugal) has is a bit of a handicap for us."
Although the French made it out of the group stage for the first time since the 2006 World Cup, the manner of Saturday's loss will be disappointing for a team touted as one of the tournament's dangerous outsiders.
France formed a deep defensive line and watched awe-struck as Spain flicked the ball around like a training session. The Spanish made dummy runs everywhere early on, and had a penalty claim waved away in the sixth minute when Cesc Fabregas tumbled under a challenge from left back Gael Clichy.
Fabregas also started against Italy in Spain's opening 1-1 draw, sparking criticism against Del Bosque for not picking a recognized striker. But against France, it worked.
Spain pulled a compact France team out of position in the 10th minute as Iniesta curled a ball behind center half Adil Rami but it was too far for Fabregas to reach.
Karim Benzema, looking for his first goal at a major tournament, looked sharp and made some good runs, but the Real Madrid striker rarely got any service. He was fouled in the 32nd after galloping onto Florent Malouda's pass, and Cabaye's free kick was sailing into the top left corner before Casillas shuffled across quickly to save it.
While the French averted a repeat of the 2010 World Cup crisis — when the entire squad went on strike at training — the buildup to Saturday's game was marred by tensions in the camp following a heated dressing room bust-up after the 2-0 loss to Sweden.
"The big teams have all showed up at this Euro, and they will be in the semifinals," Blanc said. "Unfortunately, France won't be."
