Alex Lozowski opened the scoring before Daly crossed three times in seven minutes as Sarries reached the last 16.
Marco Riccioni and Andy Christie also scored in the opening 40 minutes, while Josiah Maraku and Ethan Dumortier dotted down for Lyon at StoneX Stadium.
Dumortier and Arno Botha handed Lyon a bonus point before Ben Earl hit back.
Three-time winners Sarries were without talisman Owen Farrell, whose four-week ban for a high tackle was reduced to three weeks by the Rugby Football Union before kick off, allowing him to play in England’s Six Nations opener with Scotland on 4 February.
In a reshuffled midfield, which included versatile back Alex Goode at fly-half, outside centre Lozowksi stepped off his line to intercept an inaccurate Lyon pass to run clear for the opener.
Then it was the Daly show as the full-back threw a dummy pass and glided past defenders for his first try, before finishing team moves out wide for his second and third.
Maraku gathered the loose ball to score Lyon’s first try against the run of play but Riccioni scored the hosts’ fifth try from the back of the maul and Christie powered over for the sixth.
Dumortier crossed for the visitors on the stroke of half-time before chipping forward and claiming the bouncing ball for an excellent solo try minutes after the interval to hand the French side hope.
Botha stepped inside to score the visitors’ fourth try for the bonus point as Lyon moved to within 13 points but Earl had the final say by grounding the ball at the back of the maul with the clock in the red.
Saracens: Daly, Malins, Lozowski, Tompkins, Maitland, Goode, Van Zyl; Mako Vunipola, George (capt), Riccioni, Itoje, Tizard, Christie, Earl, B Vunipola.
Replacements: Woolstencroft, Mawi, Judge, Isiekwe, Dan, Davies, Manu Vunipola, Lewington.
Lyon: Dumortier; Tuisova, Maraku, Godwin, Veredamu; Smith, Pelissie; Kaabeche, Marchand, Gomez-Kodela, Kpoku, Mayanavanua, Cretin, Sobela, Botha.
Replacements: Coltman, Taofifenua, Bamba, William, Goujon, Saginadze, Doussain, Arnold.