After a creditable 0-0 first-leg draw in Oswestry, the Welsh Premier League side harboured hopes of securing a famous result in Sweden.
But the visitors fell behind after just eight minutes, Walid Atta netting for the Allsvenska side.
Thomas Sorum made it 2-0 before half-time and Alvaro added a late third to send TNS out of the competition and Helsingborgs through to the third qualifying round.
There was disappointment in Northern Ireland too, where Linfield drew 0-0 with AEL Limassol of Cyprus to lose 3-0 on aggregate.
The night's results in full:
Linfield (Northern Ireland) 0-0 AEL Limassol (Cyprus)
AEL Limassol win 3-0 on aggregate
Dinamo Zagreb (Croatia) 3-2 Ludogorets (Bulgaria)
Dinamo Zagreb win 4-3 on aggregate
Slask Wroclaw (Poland) 0-1 FK Buducnost Podgorica (Montenegro)
Slask Wroclaw win 2-1 on aggregate
Vardar Skopje (Macedonia FYR) 0-0 BATE Borisov (Belarus)
BATE Borisov win 3-2 on aggregate
Helsingborgs (Sweden) 3-0 The New Saints (Wales)
Helsingborgs win 3-0 on aggregate