Jason Pominville, Paul Gaustad and Tim Connolly also scored for the Sabres, who bounced back from a one-goal loss in Pittsburgh on Friday to improve to 10-4-1 in their past 15 games.
Miller made his 28th consecutive start for Buffalo, which won its seventh straight at home against Toronto.
Nikolai Kulemin and Francois Beauchemin scored, and rookie James Reimer allowed five goals on 30 shots before being yanked after two periods for the Maple Leafs, who had their two-game winning streak snapped.
Reimer was coming off his first career NHL shutout on Thursday against Carolina. He was relieved by Jean-Sebastien Giguere, who gave up a goal on 13 shots over the final 20 minutes.
Toronto lost for the fifth consecutive time on the road after winning a season-high five in a row away from home.
Down 5-1 after two periods, the Maple Leafs cut it to 5-2 on Beauchemin's goal with 7:21 left.
Vanek capped the scoring with his team-leading 19th of the season with 3:17 to go.
Leading by a pair of goals after the first period, Buffalo pushed its lead to 3-0 on Gaustad's seventh of the season when he snapped in a shot from the right circle at 4:19 of the second.
Connolly then made it 4-0 with a power-play goal just under 3 minutes later before Kulemin scored Toronto's first goal of the game with a short-handed tally at 8:50 of the second.
Stafford scored his second of the game, and 17th of the season, with 5:38 left in the second with a wrist shot from the high slot to make it 5-1.
The Sabres struck first just 3:31 into the game when Pominville netted his 12th of the season. After getting a backhand pass from Vanek in the slot, Pominville wristed a shot into the top-right corner to give the Sabres a 1-0 lead.
Stafford made it 2-0 on the power play just under 6 minutes in. He one-timed Jordan Leopold's cross-ice pass from the left circle.
Notes: The Sabres improved to 29-9-1 against Toronto since 2005-06. ... The most recent Maple Leafs win in Buffalo was Dec. 12, 2008. ... The Sabres have killed off 37 straight penalties against Toronto dating back to March 2009. ... Miller's career high for consecutive starts is 34, set in the 2007-08 season. ... Buffalo outshot Toronto by an 11-1 margin over the first 10 minutes of the first period, but were outshot 9-1 over the final 10 minutes of the frame.