Jamie McGinn also scored and Louis Domingue made 14 of his 31 saves in the third period for the Coyotes (22-32-7).
"After the first goal, Louis was really solid and it's a credit to the players that they worked hard tonight," Arizona coach Dave Tippett said. "[Vrbata] is a smart player who's had a really good season. For a guy who wasn't sure how it was going to go (at age 35) and we weren't sure how it was going to go, he's come in and played very well. He's a consummate pro who shows up and does his job every night."
Jack Eichel and Evan Rodrigues scored for the Sabres (26-26-10). Nilsson made 26 saves after Robin Lehner was injured during a 5-3 loss against the Colorado Avalanche on Saturday.
Buffalo lost back-to-back games against 30th-place Colorado and 29th-place Arizona and missed a chance to pull within five points of the Toronto Maple Leafs for the second wild card into the Stanley Cup Playoffs from the Eastern Conference.
"It's tough to swallow," said Eichel, who extended his point streak to six games with his 14th goal at 18:01 of the first period. "We've blown a lot of leads this year, and that's one of the most frustrating things. It seems like we play good and we get the lead and then we decide we're just going to sit back. … I don't know if we played a little too passive, but we don't push for that next goal, that third goal."
Rodrigues made it 2-0 at 9:53 of the second when he finished off a 4-on-2 break and beat Domingue over the blocker for his second goal in as many games.
But Arizona tied it with two goals seven minutes apart in the third period. Vrbata deflected Rasmus Ristolainen's clearing attempt to Domi, who beat Nilsson with a one-timer between the circles at 6:42.
"We turned the puck over there and gave them a freebie from the slot," Sabres coach Dan Byslma said. "After that, we let them come at us too hard. We were passive in the neutral zone and they came at us with a ton of speed."
McGinn tied it at 13:41, burying a bouncing puck past Nilsson at 13:41 for his ninth goal of the season and 100th of his NHL career.