The hosts had the better of the first half as Real began slowly, with Eduardo Camavinga almost conceding a calamitous own goal when he blocked Antonio Rudiger's clearance before Thomas Partey shot straight at goalkeeper Thibault Courtois.
Real, bidding for a record-extending 16th title in the competition, grew into the contest and threatened on the break, though their few opportunities came from Arsenal errors and the visitors looked disjointed throughout the contest.
Courtois was by far the busier keeper and produced a stunning double save, the first from a powerful Rice header, on the stroke of halftime.
There was no stopping Rice's opener, though, a fierce curling shot around the wall and beyond Courtois in the 58th minute for his first senior goal from a direct free kick.
"It's been in the locker, but I've hit the wall too many times or it's gone over the bar. Originally I was going to cross it and then I've just have seen the wall with the goalkeeper's position. I thought ... just go for it," Rice said.
Arsenal could have been 2-0 up shortly after but for more heroics from Courtois, who produced a diving stop from Gabriel Martinelli followed by another from makeshift striker Merino in between David Alaba's goal-line block.
Rice's second free kick was even better than the first, curled straight into the top corner in the 70th minute, to leave the 15-times European champions looking stunned.
Merino then coolly fired Myles Lewis-Skelly's pull-back into the bottom corner 15 minutes from time, following a pass from substitute Leandro Trossard, to give Arsenal a sizeable lead to take to Madrid next week.
The winners of the tie will face either Aston Villa or Paris St Germain, who meet on Wednesday, in the semifinals.
'BRILLIANT GAME'
"We've had a brilliant game, we had a lot of motivation, that's why we won," Arsenal manager Mikel Arteta said. "We have a game next week in the Bernabeu, we have to demonstrate we can do that again next week."
Real will have to try and overhaul Arsenal's lead without Camavinga, who was shown a second yellow card in the closing seconds for kicking the ball away.
The visitors did have some half-chances, most notably through Kylian Mbappe who scuffed his shot at the end of a fast break with the score at 0-0 in the first half.
But Real were unconvincing in attack and have now conceded 11 goals in their last four games in all competitions, while Arsenal have not lost a European tie when scoring three goals in the first leg.
Unused substitute Lucas Vazquez conceded Real were nowhere near their best but nonetheless, given his side's European pedigree, warned Arsenal his team were far from finished.
"The tie is complicated, but if there's one team in the world that can turn it around, it's us, in our stadium and with our fans," he told Movistar Plus.
Real will hope to take heart from the 1975-76 European Cup when they lost 4-1 away to Derby County in the first leg of the second round but won the return 5-1 at home.
Late goal gives Inter advantage
Inter Milan scored an 88th minute winner through Davide Frattesi to snatch a 2-1 victory over shocked hosts Bayern Munich in this morning's other quarterfinal.
Frattesi tapped in from close range to stun the Bavarians, who had levelled three minutes earlier through substitute Thomas Mueller, a player leaving the club at the end of the season after 25 years as Bayern did not offer him a new contract.
Inter captain Lautaro Martinez had put the visitors in the driving seat after 38 minutes, drilling the ball into the net from a superb Marcus Thuram backheel after Bayern had earlier hit the post through Harry Kane.
The treble-chasing Italians, who have conceded three goals in their 11 Champions League matches so far this season, host Bayern in the return leg on April 16.
"Scoring in the 88th minute like that shows how much we wanted to keep playing our football and using the principles we’ve been relying on for almost four years now," said Inter coach Simone Inzaghi.
The winners of the quarterfinal will face either Borussia Dortmund or Barcelona, who play their first leg on tomorrow morning.
The Germans, who were missing several key players to injury including goalkeeper Manuel Neuer and attacking midfielder Jamal Musiala along with a trio of defenders, had two chances through Michael Olise and Kane in the opening 15 minutes.
They came even closer in the 26th after Olise outplayed almost the entire Inter defence to lay the ball off for Kane whose curled shot hit the post.
The Italians gradually grew bolder with Carlos Augusto firing into the side netting and Martinez charging into the box but slipping just before shooting.
The Inter captain, however, made no mistake when Thuram found him with a backheel flick from a Carlos Augusto cut-back and the Argentine completed the swift move with a deft finish.
He came close to adding another in 55th but Bayern keeper Jonas Urbig stopped his powerful drive from a tight angle.
Bayern, who lead the Bundesliga title race and face Borussia Dortmund on Saturday pinned Inter in their own half but neither Konrad Laimer, Raphael Guerreiro nor Kane with a low drive in the 80th could find an equaliser until crowd favourite Mueller was in the right place to score with five minutes left.
Their joy, however, only lasted three minutes with Frattesi tapping in the winner in a heartbreaking finale for the Bayern fans in the Allianz Arena, site of this year's final.
"It's halftime, it's 2-1 for Inter and we need to go to Milan to win," said Bayern coach Vincent Kompany. "We have lost and we had our chances in both halves but the results are not always fair."
"If you see how many chances we had against a defensive-minded Inter. Normally if we get these chances we score two or three goals," added Kompany. "We still fully believe in our chances in Milan."
