Glasgow went behind when a defensive lapse allowed Richard Haughton to sneak in and win the race to reach Glen Jackson's lob as it trundled across the line.
Irish World Cup referee Alan Lewis awarded the try on the say-so of video assistant Brian Fitzgerald - despite compelling evidence the ball was marginally out of play.
Dan Parks eased Glasgow's jitters when he slotted a penalty, but Jackson restored his side's advantage when the hosts were guilty of holding on to the ball in contact.
The penalty ping-pong continued when Parks struck again from close range while Jackson added his second goal before the break to give his team more breathing space.
The half ended in dramatic fashion as first Parks completed his penalty treble, then Sarries pounced for a second try as Jackson plucked Johnnie Beattie's pass out of the air for a simple score which he converted to open up a nine-point gap.
Glasgow refused to buckle and the tie was back in the balance thanks to a well-worked try by Bernie Stortoni. Kevin Tkachuk and Thom Evans combined slickly to create the space for the Argentinian to finish in style.
Their prospects of a fightback were dashed though when Jackson kicked another penalty - though Parks added to his tally five minutes from time to salvage a consolation bonus point.
In Friday's other match, Biarritz ensured Viadana remained winless in Pool 4 as they ran out 25-16 winners at the Parc des Sports Aguilera.
USA winger Takudzwa Ngwenya ran in two of the hosts' four tries, with Jacques Cronje and Manuel Carizza adding the other touchdowns and Marcelo Bosch kicking one conversion and a penalty.
Calvin Howarth was the star for the Italians, scoring one converted try and landing three conversions.