Everton claimed that Dennis Bergkamp fouled David Unsworth before Vieira shot, but referee Alan Wiley allowed the goal.
It was fitting that Vieira should score the decider. He was the driving force behind Arsenal's rally after Wayne Rooney struck a brilliant solo equaliser.
The Gunners skipper said: "We were delighted to take three points because the pressure was right on us after United's win yesterday. It could be one of the most important goals I've scored and I don't score too many."
He added: "We were very disappointed to go out of the Champions League on Wednesday, but we responded. Some people were writing us off for the season and we answered them."
Arsenal grabbed an eighth minute lead when Pascal Cygan arrived unmarked to head home Thierry Henry's in-swinging corner.
They created more chances to build a comfortable lead. Everton keeper Richard Wright saved at Henry's feet, the Frenchman shot wide from Bergkamp's pass and Wright denied him again from Robert Pires's sweeping pass.
But Rooney, a growing influence after a slow start, exploited the defensive frailties on Arsenal's left side early in the second half, running at Cygan and unleashing a pinpoint low shot that gave Stuart Taylor no chance.
Arsenal never found their rhythm against hard-working Everton. But Vieira's strike was the sort of 'ugly' goal that Arsene Wenger had mused about after defeat in Valencia.
Everton flung on the towering Duncan Ferguson and fired long balls into the Gunners box, but Sol Campbell proved a solid barrier.
The game finished with Arsenal running the ball to the corners to play out four minutes of stoppage time. It was hardly their usual attractive game, but the home fans' roars of relief at the end epitomised the edginess around Highbury.