Eichel led the way with a goal and an assist for the Sabres, while Sam Reinhart tallied a pair of helpers. O'Reilly and Zemgus Girgensons each scored a goal, while Lehner made 29 saves on 33 shots.
Rickard Rakell, Ondrej Kase and Corey Perry scored goals for Anaheim. Ryan Miller made 30 saves.
Henrique's goal was preceded by an errant pass on the part of Ristolainen, who was unable to connect with Eichel as he sped through the neutral zone in overtime. The missed opportunity resulted in a crucial change of possession, as Henrique scored the winning goal coming back the other way.
Ristolainen's missed pass may have been the most memorable, but Eichel said it was lackluster execution throughout the game that cost the Sabres in an otherwise strong performance.
"I thought we did good things, but you battle back against a desperate hockey team and obviously you want the two points," he said. "I think there's the opportunity to make a few plays and we're just failing to make them. I think that's been the tale of our season."
Execution hurt the Sabres particularly in the second period, which they began with a 1-0 lead thanks to a shorthanded goal from Girgensons. Rakell scored a power-play goal on a rebound to tie the game just 21 seconds into the period, and the Sabres were outshot 14-5 until the second intermission.
Buffalo nearly escaped the period with a 2-1 lead despite the lopsided shot totals, thanks to a power-play goal scored by Eichel with 2:42 remaining. The Ducks answered, however, by scoring again on a rebound, this time off the stick of Kase in the final minute of the period.