Suarez scored against his former club in the first half but Jurgen Klopp's men were in relative control in the second half until Messi struck twice in seven minutes to change the entire complexion of the tie.
Messi's first came via a couple of huge slices of luck as he was able to walk the ball into an empty net, but his second and landmark goal was a fabulous long-range free kick that sailed into the top corner of Alisson Becker's goal.
Liverpool had plenty of chances themselves, most notably Sadio Mane in the first half and Mohamed Salah hitting the post with a simple finish after substitute Roberto Firmino had an effort cleared off the line.
It was not to be for the visitors though, and they now must produce a miracle at Anfield next week to somehow claw their way into the Champions League final - although Messi and Barcelona look nailed-on to be playing in Madrid on June 1.
Liverpool suffered a blow in the 21st minute when Naby Keita limped off with an apparent groin problem, being replaced by Jordan Henderson, and they were still rearranging when Barca broke the deadlock five minutes later Suarez superbly poked home Jordi Alba's excellent cross.
Mane missed a great chance to level for Liverpool soon after, but it was in the second half that Liverpool really took hold of the game, with James Milner having a curling effort turned away by Marc-Andre Ter Stegen, before the keeper got down well to steer Salah's low effort around the post.
Milner wasted a glorious opportunity when firing straight at Marc-Andre Ter Stegen from 12 yards out as the Reds piled on more pressure, but they were soon left to rue those missed opportunities.
A catalogue of lucky bounces all went Barca's way as before Suarez kneed the ball onto the bar and the rebound fell kindly to Messi to control the ball and walk it into the empty net.
Soon after the game and perhaps the tie was over as Messi lined up a 35-yard free kick and curled the ball right into the top corner - with maybe a slight question mark over Alisson's positioning and the amount of p;layers in the wall but it could not have been placed any better.
Liverpool still kept on coming though and could have grabbed a vital away goal right away when Firmino slotted under Ter Stegen but saw the ball cleared off the line.
Salah was on hand for the rebound but his right-footed effort hit the post when he really should have scored.
Henderson headed wide in another late Liverpool chance, while Barcelona wasted a couple of chances to make it 4-0 on the break as it finished 3-0 to put the Spanish champions on the brink of a return to the Champions League final.
Barcelona 3-0 Liverpool- Opta stats
- Lionel Messi scored his 600th Barcelona goal in all competitions, 14 years to the day since he scored his first against Albacete in May 2005.
- Barcelona extended their record Champions League run of 32 home matches without defeat (W29 D3 L0), with this their first ever home win over Liverpool in European competition at the fifth attempt (W1 D2 L2).
- Liverpool suffered their joint-heaviest Champions League defeat, also losing 0-3 to Real Madrid in October 2014.
- Liverpool have lost the away leg of their last six major European semi-finals, four of which have been in the Champions League (previously 2006-07, 2007-08 and 2017-18).
- Barcelona (502) became just the second team to score 500 Champions League goals, after Real Madrid (551).
- Barcelona’s Luis Suárez scored his first Champions League goal of the season, with his 36th shot of the campaign; his previous goal in the competition was in April 2018 against Roma, whose goalkeeper that day was Alisson.
- Barcelona’s Lionel Messi has scored 26 Champions League goals against English clubs – 14 more than any other player.
- Liverpool have now lost more matches in the Champions League this season (4 in 11 matches) than in all other competitions combined (3 in 38 matches).
- Only Raúl (33) has scored against more different Champions League opponents than Barcelona’s Lionel Messi (32), scoring his first goal in his third appearance against Liverpool.
- In the top five European leagues in all competitions, Barcelona’s Lionel Messi has scored eight direct free-kick goals this season – twice as many as any other player.