City remains fourth on 46 points, with Liverpool on 45 and ahead of Villa only on goals scored. Tottenham Hotspur, away to Wigan Athletic later on Sunday, have 43 points and will go fourth with a win over the team they beat 9-1 earlier in the season.
Chelsea are top on 61 points after their 2-0 win at Wolverhampton Wanderers on Saturday, taking full advantage of Manchester United's surprise 3-1 defeat at Everton which left the Premier League champions on 57.
Arsenal are third on 55 following their 2-0 home win over Sunderland.
The battle for fourth place, and the final Champions League qualifying berth, was to the fore on Sunday and Villa was first up at home to second-bottom Burnley.
The hosts trailed to an early Steven Fletcher goal as Burnley dreamed of a first away win of the season and though Ashley Young leveled after 32 minutes, it was still tense 10 minutes into the second half.
Villa then produced a spell of terrific attacking and exploded with four goals in 12 minutes. Stewart Downing got the first two and Emile Heskey and Gabriel Agbonlahor also weighed in as Burnley's defense lost concentration.
Martin O'Neill's Villa, who face Manchester United in the League Cup final next Sunday, then eased off the gas but Martin Paterson's late score for the visitors was no consolation.
The eagerly awaited battle at Eastlands turned out to be one of the dullest games of the season, as City and Liverpool seemed content to play for a draw from the start.
An appalling first half did not feature a chance until Liverpool's Martin Skrtel headed wide in stoppage time.
The second half was little better, with Pepe Reina's save to deny City's Emmanuel Adebayor the only notable action.
Liverpool boss Rafa Benitez eventually introduced Yossi Benayoun and Fernando Torres for the last quarter but even they could not lift the torpor.
In other games on Sunday Blackburn Rovers beat Bolton Wanderers 3-0 in a Lancashire snowstorm with goals by Nikola Kalinic, Jason Roberts and Gael Givet while a late free kick by Bobby Zamora gave Fulham a 2-1 home win over Birmingham City.