PARIS — France, Germany and Belgium sealed their places in the quarterfinals of the European Championships on Sunday as three of the tournament favorites lived up to their promise.

There was a scare for the host nation after Ireland scored a second minute penalty, but Antoine Griezmann struck twice after the break to give France a deserved 2-1 victory in Lyon.

While France was made to work hard to advance, Germany had no such difficulties in a polished 3-0 victory over Slovakia, and Eden Hazard inspired Belgium to what ended up as a crushing 4-0 win over Hungary.

The goals and the quality of the football over the three games were a marked improvement on the dire performances witnessed Saturday when Poland, Wales and Portugal advanced.

The relief was palpable at the Stade de Lyon after France finally overwhelmed an Ireland side that had led through Robbie Brady’s spot-kick, after France midfielder Paul Pogba bundled Shane Long to the ground.

Ireland chased for every ball and worked tirelessly to defend its slender lead. But Griezmann’s two goals in the space of three minutes in the second half saw the natural order restored, and a red card for Shane Duffy in the 66th minute all but ended the contest.

France’s reward is a quarterfinal next Sunday at the Stade de France in Saint-Denis against either England or Iceland, who play Monday.

Germany scored early and never looked back against a Slovakia side that had surprisingly beaten Joachim Loew’s world champions in a friendly last month. Jerome Boateng broke the deadlock after just eight minutes, and Mario Gomez scored just before the break as Germany advance to a matchup against Spain or Italy.

In Toulouse, Toby Alderweireld headed Belgium into a 10th-minute lead, and Hazard sparked a late surge with an assist and a goal as Belgium lined up a quarterfinal against Wales.