Brendan Rodgers' side dominated the second leg of their qualifying round play-off at Anfield but were punished for not adding to their 1-0 aggregate lead when Reina let a David Templeton shot squeeze through late on.
With extra time looming, Liverpool rallied and Suarez made amends for missing two earlier gilt-edged chances with a fine finish in the 88th minute.
It was a victory the Reds thoroughly deserved, played out against the backdrop of Friday's impending transfer deadline.
A Motherwell team featuring four teenagers in the starting line-up held Levante for 72 minutes in Spain before their European campaign ended in another defeat.
Michael Higdon had a goal disallowed as an injury-hit Motherwell gave as good as they got in the first half, before Greece striker Theofanis Gekas finished off their Europa League hopes as he converted a cross to complete a 3-0 aggregate win and ensure Levante progress to the group stages.
Motherwell had disappointed in the home leg but they restored pride tonight against a Levante side who beat Real Madrid last season and topped La Liga in late October.
Inter Milan twice came from behind as they claimed a 2-2 draw at home to Vaslui and a place in the group stage of the Europa League.
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Football Transfer CentreThe latest football newsFollow us on TwitterThe Italians held a 2-0 lead from the first leg, but trailed to Nicolae Stanciu's 36th-minute penalty.
Rodrigo Palacio equalised after 76 minutes but Vaslui were in front again three minutes later through Fernando Varela.
However, a last-gasp goal from Fredy Guarin spared Inter's blushes at the San Siro and wrapped up a 4-2 aggregate victory.
Serie A rivals Lazio had a much more straightforward time of things against Mura, winning 3-1 to seal a 5-1 victory overall.
Libor Kozak scored twice while Mauro Zarate was also on target as the home side clocked three goals before the interval. Mura's only response came from the boot of Jure Travner late on.
Athletic Bilbao could only manage a 3-3 draw at HJK Helsinki, but a 6-0 win in the opening leg saw them through to the group stage 9-3 on aggregate.
Mikel San Jose, an own goal from Rami Hakanpaa and Igor Martinez's late strike dug Athletic out of a hole after the Spanish side had trailed 2-0 and 3-1.
PSV Eindhoven had not concerns in their play-off against Zeta, winning the second leg 9-0 at home to complete a 14-0 aggregate scoreline.
Tim Matavz and Georginio Wijnaldum both scored hat-tricks for the Eredivisie side, with Peter van Ooijen (two) and Mathias Jorgensen also netting.
FC Twente needed an extra-time goal from Leroy Fer to see off the challenge of Bursaspor.
The Turkish side had won the first leg 3-1, but Twente notched up the same score in the return match to take the side into extra-time.
Fer's late heroics then wrapped up a 5-4 aggregate success.
Heerenveen saw their Europa League challenge ended by Molde, who won 2-1 on the night and 4-1 on aggregate, while AZ Alkmaar suffered a humiliating 5-0 defeat at home to Anzhi Makhachkala to lose to tie 6-0.
Feyenoord were also dumped out of the competition, suffering a 2-0 defeat at Sparta Prague and a 4-2 loss on aggregate.
Marseille could only manage a goalless draw at home to Sheriff on Thursday night but their 2-1 win from the first leg was enough to carry them through.
An injury-time penalty from Jussi saw Bordeaux edge out Crvena Zvezda 3-2 on the night and by the same score overall.
Hannover beat Slask Wroclaw 5-1 to seal a comprehensive 10-4 victory.
Maritimo, Dnipro, Neftchi, AIK, Hapoel Tel Aviv, Rosenborg, Steaua Bucharest, Young Boys, Metalist Kharkiv, Genk, Plzen, Club Brugge, Partizan Belgrade, Rapid Vienna, Videoton and Sporting Lisbon also ensured their places in Friday's draw.