Smith had surgery to repair an abdominal injury and missed 40 games. The shutout was his second of the season and his 29th in the NHL. He is 12-1-1 against the Oilers.
"It was a tough turnaround from last night (a 4-1 win at the Calgary Flames) and we knew the goaltender had to be good for us tonight," Coyotes coach Dave Tippett said. "He was anxious to come back. On a tough turnaround, we put him in a tough situation and he reacted and played very, very well."
Cam Talbot made 25 saves for the Oilers (27-37-7), who were shut out for the second time in three games and seventh time this season.
"The smarter team won the game, hands down," Oilers coach Todd McLellan said. "We had 44 shots on net, but other than [Taylor Hall] breakaway, I don't recall having one single outnumbered rush. On the other end, we gave up five or six clear-cut 2-on-1s. Reckless pinches, no coverage, we gave up a breakaway on our power play where [Martin] Hanzal walked in all by himself."