The competition has already produced upsets this week with Premier League sides being knocked out by lower league opposition and Everton came close to being another victim.
The Toffees were pushed all the way by their League One opponents but advanced to the third round after two late goals gave them a 4-2 victory at Sincil Bank.
Headers from Alex Iwobi and Richarlison in the final 10 minutes finally swung the tie decisively in Everton's favour.
Harry Anderson had put the hosts in front after only 21 seconds before a superb Lucas Digne free-kick and Gylfi Sigurdsson's penalty turned the game on its head.
However, a thunderous volley from Bruno Andrade levelled things up again before the late drama.
Silva showed how seriously he is taking the competition after putting out a strong side and was relieved to see them fight their way through.
Moise Keane was one of four players to make their full debut.
The striker, signed from Juventus this month, was one of five changes to the team as fellow summer recruits Fabian Delph, Iwobi and Djibril Sidibe also made their first starts for the Blues.
The other change from the Premier League defeat at Aston Villa saw Mason Holgate selected in defence.
"It was important that we were serious and respected the competition and respected Lincoln," said Silva.
"We started in the worst way but after that we did our best to stay in the competition. We were losing but we were playing the way we wanted to play.
"We scored a great goal with Luca but even before that, we had created a lot of chances and we were doing the right things.
"We showed personality to keep playing the way we should play and we did some things really well. We did our obligation and we deserved to go through to the next round."
The third-round draw took place after Everton's win at Sincil Bank, with Silva's side facing a trip to Hillsborough to play Sheffield Wednesday.
It will be a tough encounter for the Toffees but Silva has a cup run on his mind and will arrive in south Yorkshire with a strong side determined to go through.
"It will be another tough match," he added. "Sheffield Wednesday is a Championship club who have started well and we will do our best to stay in the competition."