The proposal was approved by 48 votes to six with no abstentions at a meeting of Parliament's civil liberties committee on Tuesday.
Monica Macovei, who is Parliament's rapporteur on the dossier, defended the introduction of systematic checks.
The ECR group MEP said that checking all EU citizens, as well as family members with third country nationalities, against EU-wide and national internal security databases, or those listing lost or stolen travel documents, would make it easier to apprehend travellers trying to hide their real identities.
The draft regulation is a response to the rise in terrorist threats in Europe, such as the recent attacks in Paris, Copenhagen and Brussels. It also aims to combat terrorist foreign fighters - many of whom are EU citizens - irregular migration and human trafficking.
An estimated 5000 EU citizens have travelled to conflict zones and joined terrorist groups, such as Islamic State.
Speaking at a news conference after the committee vote, Macovei said, "The right to life is the most basic of human rights. The recent terrorist attacks bitterly demonstrate the current threat to Europe's internal security and proves that changes are needed. It is also worth giving up some of our comforts and time, even if this means longer queues at the borders, in order to save lives."
She said it was hoped that the proposal would help deter those who "want to do evil."
"We cannot wait for the next terrorist attack. We have to take preventative measures. We have to accept that the public has lost confidence in the EU, partly because of the perceived inability to counter the terrorist threat."
Committee members amended the Commission proposal to enable member states to run "targeted" checks, as an exception, in the event of lengthy delays and provided that security is not at risk.
Under the proposal, checks will be made against various databases, including Interpol's stolen and lost travel documents database (SLTD) and the Schengen information system.
The proposal, which amends the current Schengen borders code, would introduce more thorough checks than the current ones at all EU external air, sea and land borders.
However, Macovei said that if systematic controls slow down border traffic flows too much, the EU member states should be able to carry out targeted checks at specified border crossings instead.
This would only happen, she added, if it first can be concluded that such a relaxation of the rules would not increase security risks.
"Every risk assessment should also be based on EU-wide security indicators developed by the Commission, the new EU border and coast guard agency and the Council," says the text adopted by the committee.
Each analysis would have to be sent to the border member states' relevant authorities, the EU border agency and the Commission, it adds.
Better data management, technological progress and improved connections between member states' information systems should nevertheless ensure that the checks have a limited effect on the duration of borders crossings, said Macovei.
Where targeted checks are introduced, all travellers would still have to undergo a minimum, rapid check on the standard travel documents. Biometric identifiers in passports, such as fingerprints or facial images, should also be thoroughly checked and verified whenever a person's identity is in doubt, she said.
The proposal also makes it mandatory to check all third-country nationals who are leaving the EU against relevant databases, to ensure that they pose no security threat.
MEPs will now start negotiations with Council very soon to try to reach a first reading agreement.
Macovei said, "We aim to start these talks later this week. The sooner this is implemented the better."