England international Abraham put Roma ahead in the 54th minute with a smart turn and low drive which calmed nerves among the smattering of home fans at the Stadio Olimpico
Stefano Sensi crashed the hosts back ahead with a powerful drive in the 104th minute which finally settled the match in Inter's favour.
Massimiliano Allegri's side will take on either Sassuolo or Cagliari, who face each other on Wednesday, in the last eight
Shevchenko's Genoa desperately needed a result with their only success in the league back mid-September against fellow strugglers Cagliari.
The two were born just a few months apart and they were known as 'I gemelli del gol', the goal twins, during eight glorious years together at Sampdoria.
The dethroned league champions beat Atalanta 2-1 to win the Italian Cup for the 14th time after winning the Super Cup in January.
It is a fifth league title for Conte who won three with Juventus between 2012 and 2014 and one with Chelsea in 2017.
Antonio Conte's side clinched their 19th 'Scudetto' after nearest rivals Atalanta drew 1-1 at Sassuolo to leave the 'Nerazzurri' 13 points clear with four games remaining.
Coppa Italia: Inter Milan coach Antonio Conte apologises for 'reacting badly to provocation' in Juventus defeat
Ex-Juve boss Conte appeared to show the finger to his former club's directors at half-time, while footage from broadcaster RAI at the end of the match showed the hosts' chairman Andrea Agnelli apparently telling Conte to "shut up".
Pessina's efforts, both set up by Duvan Zapata, were enough to secure Atalanta a second final in three seasons, after the goalless first leg, where they will face Juventus.
Andrea Pirlo's side will face one of Napoli or Atalanta, who are locked at 0-0 ahead of Wednesday's second leg in Bergamo, in May's final in Rome.
European football matchday: Manchester United face West Ham test in FA Cup; Juventus vs Inter Milan in Coppa Italia
Real Madrid would aim to gain momentum when they host city rival Getafe in a LaLiga match postponed from the first round.
Napoli goalkeeper David Ospina proved decisive on the night pulling off key saves as Atalanta came closest to breaking the deadlock in the Diego Armando Maradona Stadium.
Inter, who beat Juventus 2-0 last month in Serie A, have it all to do to keep their hopes of lifting a first Italian Cup since 2011 alive.
Coppa Italia: 10-man Atalanta beat Lazio to reach semi-finals; Juventus see off second-tier SPAL to reach last four
A dramatic match in Bergamo saw five different goalscorers, a red card and a missed penalty as Atalanta came out on top in a rematch of the 2019 final, won 2-0 by Lazio.
Coppa Italia: Zlatan Ibrahimovic accuses Romelu Lukaku of practicing 'voodo' in during clash, say reports
AC Milan forward Ibrahimovic and his Inter Milan opponent Lukaku were involved in a heated altercation at the half-time whistle in the San Siro.
Coppa Italia: Christian Eriksen's late strike sends Inter Milan into semi-finals as Zlatan Ibrahimovic sees red
The quarter-final at the San Siro was overshadowed by a row between star strikers Zlatan Ibrahimovic and Romelu Lukaku, who both scored on the night, with the Swede later sent off in the second half.
It was more misery for Roma, just a few days after an embarrassing 3-0 Serie A loss to Lazio in the capital derby.
Coppa Italia: Romelu Lukaku's late winner sees Inter beat Fiorentina to set up quarter-final clash against Milan
Inter will be the home side when it plays its city rival at the end of the month. The two Milan teams are also leading Serie A with the Rossoneri top of the league, three points ahead of second-place Inter.
Serie A: Juventus coach Maurizio Sarri plays down speculation of his exit, wants to honour his contract
Sarri has struggled to impose his own style of football, having to build a team around 35-year-old forward Cristiano Ronaldo