Sullivan scored in regulation and a shootout to help the Predators win their second straight game, 2-1 over the New York Islanders on Monday night.
"We're not that excited about our game tonight," Sullivan said. "We got away with one."
Vokoun struggled earlier this season, but Nashville coach Barry Trotz saw signs of the Vokoun of old returning.
"He was much more patient today," Trotz said. "He's starting to get into a flow."
Dunham has made three consecutive starts while usual starter Rick DiPietro recovers from a groin injury.
"Mike's been fantastic for us," Sillinger said. "He's played great in three straight games, and it's too bad he took the loss tonight."
Nashville played its first overtime game of the season, while the Islanders beat Anaheim in a shootout Wednesday night in California.
The Islanders wasted little time opening the scoring. Just as their first power play expired, Sillinger, down in the low right slot, took Mike York's pass-out and fired an uncontested backhand past Vokoun on the short side. The goal, Sillinger's second of the season and 200th of his NHL career, came at 4:38.
"I didn't even realize at first that it was 200," Sillinger said. "For all the years and all the teams I played for, it's a nice honor. I just wish it came in a win."
Dunham made a big save in the period when he snared Scott Hartnell's rising one-timer with his glove with 3:39 left.
Sullivan tied it just after the Predators' first power play ended at 10:15 of the second period on a one-timer from the left circle.
"I absolutely flubbed it," Sullivan said. "I think (Dunham) expected a harder shot, and he played it that way, but it floated off the crossbar and went in."
New York got a scare with 3:33 left in regulation when Dunham went down after taking Kimmo Timonen's slap shot off his mask. The goalie shook it off and stayed in the game.
Vokoun made his best save in overtime when he robbed Shawn Bates with his glove at the right post with 28 seconds left. In overtime, he stopped the dangerous Miroslav Satan and Viktor Kozlov.
"Satan made a great move," Vokoun said. "I think he felt confident he was going to get it by me. He was going to stretch my leg, and I was faked out for a second, but it hit my toe and stayed out."
- New York has only won consecutive home games at the start of the season once in their last 12 years. They beat Florida and Toronto in 2003-04.
- Nashville is one of only two teams to be scheduled for a three-game road trip against the New York area teams in one visit. Florida is the other.
- The announced attendance was 9,431.