Saturday's on-field action had played second fiddle to an emotional final tribute at Elland Road for Speed, including a minute's applause, six days after the former Leeds midfielder was found hanged at his home in Cheshire.
Leeds manager Simon Grayson - he signed on for the club as a schoolboy on the same day as Speed - and his players had managed to harness their grief in positive fashion on an emotional night at Nottingham Forest on Tuesday, turning in arguably their best display of the season to win 4-0.
Speed's widow Louise and two sons Edward and Thomas were in attendance on Saturday, together with former team-mates David Batty, Gordon Strachan and Gary McAllister as Leeds clinched their first home win in over two months.
A largely forgettable first half was followed by two moments Speed himself would have been proud of just after the hour-mark when Snodgrass secured the points with a brilliant free-kick and then, two minutes later, a brave diving header.
Grayson named the same starting XI that won so convincingly at Forest on Tuesday night, but there was not even a place on the substitutes' bench for leading goalscorer Ross McCormack.
Opposite number Kenny Jackett made two changes, recalling skipper Paul Robinson and striker Darius Henderson.
The early on-field action was overshadowed by the tributes to Speed and just as Leeds' travelling fans had done at the City Ground on Tuesday night, the home supporters began singing Speed's name in the 11th minute, non-stop for 11 minutes. At Forest, Snodgrass scored just as the tribute had stopped, but there was to be no repeat this afternoon.
Luciano Becchio headed an early corner wide, while at the other end Henderson forced Leeds goalkeeper Alex McCarthy into a super reflex save with a header from six yards.
But there was scant else to warm either set of fans and Leeds' chances of a first home win in five attempts were dealt a big blow when both captain Jonny Howson and midfield partner Michael Brown limped off injured at the end of the first half.
Lloyd Sam replaced Howson and striker Andy Keogh was sent on for Brown.
Leeds began the second half out of kilter due their enforced reshuffle and Millwall began to grow in self-belief, but Scott Barron will have felt he should have done better with his free-kick after Paddy Kisnorbo had fouled Jack Smith on the edge of the area.
Sam began to make headway down the right for Leeds though and he swung in a fine cross that Becchio headed straight at visiting goalkeeper Steve Mildenhall just before the hour.
Leeds then hit Millwall with a double blow in the space of two minutes.
Millwall defender Robinson brought down Leeds midfielder Adam Clayton 25 yards from goal and Snodgrass curled the subsequent free-kick, left-footed, sublimely into the top corner in the 62nd minute.
Two minutes later Sam whipped in another telling cross and Snodgrass got in front of his marker to send a diving header goalwards that Mildenhall could only parry into his net.
Leeds threatened to increase their lead and Mildenhall saved well to deny Becchio a third goal in the 88th minute and kept out Sam's curling effort in time added on.