Brendan Rodgers’ team had a 1-0 advantage from last week’s first leg at Tynecastle and despite dominating for long periods at Anfield, they only just scraped past the Scottish Premier League outfit.
David Templeton looked set to make Liverpool pay for their wastefulness when his weak shot was allowed into the net by Pepe Reina with five minutes left.
But three minutes later Suarez burst away down the left and managed to find the net from close-range to secure Liverpool’s place in the group stages of Europe’s second tier competition.
It was an unnecessarily nervous end to the evening for a Liverpool side that had plenty of chances to see off the Edinburgh club.
“We knew it was going to be tough,” Liverpool’s Stewart Downing said on ESPN. “We made more hard work of it than we should have done. We were a little bit wasteful in front of goal.
“However, the manager stressed to the team ‘don’t go down under pressure’, and I think we bounced back well.
“We created lots of chances we didn’t finish though and we have to be more clinical in front of goal. That was a lesson for us.”
Rodgers, who has allowed Andy Carroll to join West Ham on a season-long loan, can also be forgiven for having half an eye on the 24 hours ahead.
The Liverpool manager could see Charlie Adam and Jay Spearing leave and is be keen to add new faces ahead of Friday’s 2200GMT deadline.
Hearts began brightly and Callum Paterson hit a half volley that landed just wide and had another effort blocked by a team-mate.
Suarez immediately carried a threat and after swapping passes with Martin Kelly, he had a low effort pushed away by Hearts goalkeeper Jamie Macdonald.
Downing has endured plenty of criticism during his time at Anfield but he showed great skill to beat Paterson with ease and send over a cross that was flicked on by youngster Adam Morgan and headed towards goal by Suarez, only for Marius Zaliukas to clear off the line.
John Sutton went down in the area under Jamie Carragher’s challenge with the visitors appealing for a penalty, before Liverpool’s Joe Allen broke forward and had a long-range effort held by Macdonald.
Sutton glanced a header wide from a whipped cross from Templeton and Morgan wasted a decent opening after he was played through by Steven Gerrard after a quick break.
After the break, Gerrard picked out Martin Skrtel at the near post but the Slovakian defender’s effort was turned wide by Andy Webster.
And Jonjo Shelvey thumped another effort from 25 yards just past the top corner as the hosts continued to press.
Suarez then passed up two opportunities to kill the tie. The Uruguayan was sent through by Jordan Henderson and had his close-range effort blocked on to the post by Ryan McGowan.
A minute later he spun 10 yards from goal and curled a shot past the far post.
Raheem Sterling, who had replaced Morgan just after the hour, made an immediate impression and just missed after an beating several defenders and Gerrard also hit a bending shot narrowly wide midway through the second half.
Hearts went close again through Mehdi Taouil and Daniel Grainger also had a free-kick deflected wide by the wall.
The Scottish side pulled the tie level in the 85th minute when Reina allowed a routine shot from Templeton to squirm through his body and into the net.
But, with extra-time looming, Suarez broke clear down the left and managed to squeeze a shot inside the near post in the 88th minute.