The Blackhawks have won five of their last six in this year's playoffs and will try to seize a 2-0 lead when they host the Red Wings on Saturday.
Oduya put a wrist shot past Howard and Marcus Krueger scored on a backhander less than 4 minutes later. Patrick Sharp delivered an empty-net goal in the final minute.
Chicago peppered Howard with 41 shots, having gained momentum during the opening round by eliminating Minnesota in five games. The Blackhawks finished the regular season with the best record in the NHL.
Crawford made 20 saves to get the win, boosting his career playoff record to 10-9.
Marian Hossa gave the Blackhawks the lead with a power-play goal that came 9:03 into the first period and Detroit tied it 1:54 later on a tally from Damien Brunner.