Christy Fagan fired the men from the Republic of Ireland into an unlikely lead in the 38th minute of the second qualifying round first-leg tie, tapping beyond goalkeeper Dusan Kuciak from close range after Ian Bermingham had slid the ball into the area.
It was then a case of protecting their precious advantage for St Patrick's, but they found themselves under the cosh for much of the remainder - Legia rattling the crossbar before the interval.
The onslaught continued in the second period, but the hosts were struggling to find a way past Brendan Clarke in the St Patrick's goal.
That was until Radovic found a gap in the visiting defence to slot home in injury time.
That ensured the sides head to the Irish Republic for the second leg all square - St Patrick's with an away-goal advantage.
Romanian side Steaua Bucharest overcame a lethargic display to earn a narrow 1-0 advantage heading into the second leg of their tie against Norwegian champions Stromsgodset.
Having been second best for much of the opening period, Steaua - who featured in the group stage of last year's tournament - eventually broke the deadlock when Gabriel Iancu lofted a fine lob over Ghana international goalkeeper Adam Kwarasey.
Oyvnid Storflor went close to levelling matters late on, but there was to be no equaliser for Stromsgodset, whose former boss Ronny Deila led Celtic to a 1-0 victory over KR in Tuesday's fixtures.
The defeat was Stromsgodset's first in competitive action at home since August 2013.
Ludogorets enjoyed a run to the round of 16 in last year's UEFA Europa League, and their bid to contest Europe's premier club competition got off to the ideal start against F91 Dudelange of Luxembourg.
After Dani Abalo and Roman Bezjak had put them in front in the first half, Anicet Abel and Fabio Espinho made it 4-0 by full-time to put the Bulgarians on the brink of the third qualifying round.
In the day's remaining first-leg fixtures, Aktobe of Kazakhstan picked up a valuable away goal in a 1-0 win at Dinamo Tbilisi in Georgia, and Ventspils held Malmo to a goalless draw in Sweden.
