Keith Tkachuk and Dan Hinote also scored for the Blues, who won their second game in two nights, both on the road. Legace held off the Devils in the third period, stopping all 13 shots - including Brian Rafalski's slap shot through a screen in the final seconds.
The Blues followed Tuesday's 4-3 shootout win in Columbus with a determined effort in which they twice overcame one-goal deficits.
After a terrible start to the season, St. Louis has won eight of its last 11.
John Madden and Brian Gionta scored for the Devils, who had won three in a row.
Drake snapped a 2-2 tie by cutting to the net just as Barret Jackman wound up to shoot. Serving as an effective screen, Drake blocked Martin Brodeur's line of sight as the puck bounced off him at 9:35.
Gionta gave the Devils a 2-1 lead 19 seconds into the second as he deflected Patrik Elias' shot past Legace. The goal was Gionta's 20th, and fifth in the last three games.
New Jersey almost got another when Zach Parise hit the crossbar about a minute later. That miss proved costly as the momentum soon swung in the Blues' favour.
Jackman got the first of his two assists in the period when Hinote poked the puck past Brodeur at 2:32 to tie the score 2-2.
The Devils got the first goal of the opening period when Madden deflected home Rafalski's pass from the right boards at 4:23.
The Blues pulled even at 10:50 as Tkachuk got his fifth goal in seven games.
Notes: Devils defencemen Colin White (groin) and Johnny Oduya (shoulder) remained sidelined. ... For the Blues, centre Ryan Johnson (shoulder), defenceman Christian Backman (knee) and wing Vladimir Orszagh (knee) were out of the lineup. ... Brodeur and Legace both stretched their strings of consecutive starts to 13. ... The Blues are 4-0 against Eastern Conference teams.