A mistake by Barca's usually excellent goalkeeper Marc-Andre ter Stegen gave the home team an equalising goal that sealed the match.
Suarez put Barca in front in the 29th minute before setting up Messi on his landmark night moments later.
The Argentinian playmaker's withdrawal will be a major concern for Barca, particularly given this was Messi's first start of the season after just recovering from a niggling calf injury sustained at the start of August.
Barcelona Vice President Jordi Cardoner, however, denied the Spanish soccer club was in contact with Neymar amid reports the Brazil striker wants to leave Paris Saint-Germain.
Manchester United underwhelmed with no registered shots on target in a 0-1 loss to Barcelona in their UEFA Champions League quarter-final first leg clash. Barca's point was an own goal from Man U's Luke Shaw.
Luis Suarez and Lionel Messi scored twice in two minutes as Barcelona broke the resistance of 10-man Atletico Madrid on Saturday to go 11 points clear of their La Liga title rivals.
Barca won a penalty early in the second half when Luis Suarez was fouled and Messi converted to put the league leaders in front and the Uruguayan forward made sure of the points with a simple finish from close range in the 82nd minute.
Luis Suarez scored twice as Barcelona inflicted another defeat on Real Madrid to move within touching distance of winning a record fifth consecutive Copa del Rey.
Messi scored Barcelona's second goal against SD Eibar shortly after halftime to reach a milestone in the competition after 435 league matches.
A win against Deportivo La Coruna allows Barcelone to secure their 25th La Liga title, and first since 2016, with four games to spare.
Barcelona is only the third team in the tournament's history to have won four successive tournaments on the trot.
The pair of own goals by Daniele De Rossi and Kostas Manolas plus some sloppy defending that allowed Gerard Pique and Luis Suarez to also score left Roma on the verge of elimination.
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