Everton now have the upper hand heading into next week's second leg, despite a wretched first 30 minutes at Goodison Park.
Oleh Husyev's near-post flick gave Dynamo a deserved 14th-minute lead and the visitors only looked like extending their lead in the early stages.
Roberto Martinez's side were simply not at the races, but Lukaku - dominant in the Europa League of late - sparked them to life, first hitting the crossbar with a 25-yard free-kick before he set up Steven Naismith's 39th-minute equaliser in stunning fashion.
The Belgium international beat three players before finding Naismith, and he then put his own name on the scoresheet eight minutes from time after Danilo Silva was adjudged to have handled Leon Osman's cross.
Lukaku's spot-kick was far from perfect, but it snuck in as he bagged his seventh goal in his last four Europa League matches to give Everton victory.
Everton looked anxious at the back in the early stages, with a series of crosses causing panic and one of them - hacked away for a corner by James McCarthy - led to Dynamo's opener.
Andriy Yarmolenko's ball in from the right was superbly flicked on at the near post by Husyev, who nipped in front of McCarthy to volley home.
Everton goalkeeper Tim Howard was alert to keep out a fierce Sergiy Sydorchuk drive from 30 yards as Dynamo searched for a second, while at the other end, Oleksandr Shovkovskiy was relieved to see a Lukaku effort from distance sail over after he charged off his line and fluffed a clearance.
The restless mood at Goodison Park was summed up when a poor Antolin Alcaraz back-pass, which almost played Dieumerci Mbokani in, was met with fury from the home fans.
Everton turned the screw just before the break, though, as Lukaku's free-kick flicked the top of the crossbar and Phil Jagielka had a header cleared off the line by Silva.
And the hosts then restored parity when Lukaku beat three defenders with a storming run forward that showcased his speed and power, before he released Naismith, who curled past the onrushing Shovkovskiy from 12 yards.
The goal was just what Everton needed to lift spirits and they picked up where they left off in an impressive start to the second half.
First, Naismith sent a glancing header just wide before Luke Garbutt's fierce free-kick forced a corner. Naismith again won the header, but could only nod onto the chest of Lukaku, as Shovkovskiy rushed off his line to clear the danger.
Mbokani jumped above Jagielka and headed onto the roof of the net from eight yards for the visitors, but Everton, now controlling the game, continued to surge forward, as Arouna Kone and Gareth Barry spurned half-chances.
Dynamo offered the odd reminder of their ability, as shown when Yarmolenko shot over, but in Lukaku, Everton had the game's best player.
And after referee Carlos Velasco pointed to the spot after Silva's indiscretion, Lukaku stepped up to score from 12 yards, even if Shovkovskiy got a trailing leg to his effort, as the hosts celebrated a character-building victory.
