The most audacious came at Stamford Bridge from the visitors. Stoke midfielder Charlie Adam beat goalkeeper Thibaut Courtois from inside his own half but it couldn't prevent Chelsea winning 2-1 to strengthen its grip on top spot with eight matches to go.
"I saw the keeper off his line, had an effort and it has gone in," Adam said of the goal, which evoked memories of David Beckham's strike for Manchester United in 1996. "I always have a look at the goalkeeper because these balls move so much."
Chelsea manager Jose Mourinho rated it as a potential goal of the season, saying: "It is the goal that every top player in the world would love to score."
Adam's goal cancelled out Eden Hazard's penalty, and Chelsea's winner was more mundane, with Loic Remy slotting into an empty net in the second half.
Arsenal rose to second by overwhelming Liverpool 4-1, with a three-goal burst coming in eight minutes. And Manchester United climbed provisionally to third with a 3-1 victory that featured an exquisite volley on the turn from captain Wayne Rooney.
Manchester City was dragged down to fourth place without playing, though the champions can return to second on Monday by winning at Crystal Palace.
The sensational strikes weren't just the preserve of the leading teams. Bobby Zamora lobbed the goalkeeper successfully as Queens Park Rangers stunned 10-man West Bromwich Albion 4-1.
QPR remains in the drop zone, while Leicester is still bottom despite beating West Ham 2-1 for its first victory in three months.
Hull is level with Villa, three points above the drop zone, after losing 3-1 with 10 men to Swansea, with Bafetimbi Gomis' overhead kick the pick of his two goals.
Southampton's collapse from being a Champions League contender continued with Everton captain Phil Jagielka clinching a 1-0 win.
At Emirates Stadium, three goals in eight minutes just before halftime secured Arsenal's victory after a string of saves from goalkeeper Simon Mignolet rescued Liverpool.
Mesut Ozil and Aaron Ramsey combined with a flowing move to set up Hector Bellerin to curl the ball past Mignolet in the 37th minute.
Two minutes later Ozil bent a free kick around the wall into the net, and Alexis Sanchez resisted Kolo Toure's sliding challenge before dispatching the ball high past Mignolet before the break.
Although Liverpool captain Jordan Henderson pulled one back from the penalty spot, Emre Can was sent off in the 84th minute. Arsenal used its man advantage to ensure the victory was even more emphatic, with Olivier Giroud scoring in stoppage time.
"We were focused and clinical," Arsenal manager Arsene Wenger said. "We had the killer instinct."
The same instinct United seems to have again, following up its victory at Liverpool before the international break with success over Villa.
There was a double from Ander Herrera — a half-volley from Daley Blind's pass in the first half and a late side-footed shot — but Rooney's eye-catching goal came between them.
The captain used his outstretched left foot to bring down Angel Di Maria's cross and made a sharp turn before using his other boot to volley the ball into the net in the 79th.
United goalkeeper David de Gea made a rare mistake a minute later, allowing Christian Benteke's shot to go through his legs, but it was only a consolation for Villa.