Depay's strike at the start of the second half ensured Xavi's homecoming at Camp Nou ended in victory, but a nervy one that will have left him under no illusions about the difficulty of the task ahead.
Raul de Tomas twice struck the post late on as Espanyol spurned a flurry of late chances that really should have seen them take a draw from a game Barca dominated for an hour and then almost let slip from their grasp.
Results, though, must come quickly and this one lifts Xavi's team from ninth in the table to sixth, six points behind Atletico Madrid, who earlier needed a late Felipe header to win 1-0 at home to Osasuna after Sevilla were held to a 2-2 draw by Alaves.
There was also an undeniable uplift in mood at Camp Nou, with the 74 418 attendance almost double the 37 278 that watched Barca draw here against Alaves two weeks ago.
At his unveiling last week, Xavi had emphasised the need for "rules" and even before kick-off there were signs of a shift. The players rolled into Camp Nou under a plume of red smoke, with the squad arriving all together on the team bus.
Barcelona's XI boasted eight La Masia graduates, including Ilias Akhomach, who was handed his debut. Ilias was the second 17-year-old in Xavi's first starting line-up along with Gavi and when he was replaced at halftime, on came the 19-year-old Abde Ezzalzouli.
In the warm-up, there was a more intense exercise of attack against defence and when Depay scored it was perhaps not just relief that saw all 10 outfield players celebrating together in the corner.
FASTER FOOTBALL
The football was faster and more fun. Barcelona's players seemed liberated in the final third, where passes were flicked, scooped and backheeled. The best of them, by Gavi, led to Depay winning the penalty.
They started in a 4-3-3 with both Gavi and Ilias hugging the touchlines and stretching Espanyol's defence out wide.
As well as the additional width, there was a clear intention to hunt the ball higher up the pitch. Barca were happy to leave defenders one against one at the back if it meant more players pressing Espanyol in their own half.
Nico almost scored when he stepped into to steal a pass from goalkeeper Diego Lopez but he looked, unsuccessfully, for a penalty instead of finishing.
There were 11 shots in the first 45 minutes and plenty of chances as Depay failed to catch an early volley before Ilias blazed over from close range.
The best spell of pressure came around the half-hour mark as Nico looked certain to score from Jordi Alba's cutback but was denied and Depay was put through but his smashed shot was blocked by the chest of Lopez at the near post.
Espanyol had glimpses of their own, De Tomas the main threat on the break. He almost punished some lethargic defending by Oscar Mingueza but the deflected finish was tipped over.
Ilias was replaced by Abde at halftime and less than a minute into the second half, Barcelona had a penalty, Depay going over after superb, heeled pass by Gavi. Leandro Cabrera clearly touched the ball but the decision stood and Depay made no mistake.
Abde almost scored a brilliant second after darting between two Espanyol defenders but without the cushion, Espanyol were always in the game and a flurry of chances followed.
A quick free-kick over the top again caught out Mingueza but De Tomas fired wide, before three openings came in five minutes.
De Tomas' free-kick drew gasps when it shaved the outside of the post. Fran Merida was free three yards out but somehow headed wide. De Tomas headed towards goal, leaving goalkeeper Marc-Andre ter Stegen rooted, but again the ball hit the post.
Xavi threw on central defender Ronald Araujo for extra solidity. In the 96th minute, Wu Lei threw one last foot at a loose ball in the area but Ter Stegen, and Barca, held on. (AFP)

AFN VP pledges N1.6m reward for C’wealth Games trials
Joshua warns Prenga ahead July 25 bout
Nigeria’s World Cup absence a big miss – Terry
NOC To Formally Launch Safeguarding Policy On June 16
Messi scores on injury return as Argentina beat Iceland in World Cup warm-up
Iwobi to join Yobo, Enyeama, Musa on 100 Nigeria caps
Omewa happy to open Falcons goal account
World Cup: Enyimba, Remo to benefit from FIFA’s $355m windfall
Nigerian-Born Stars Set to Shine in the US, Canada and Mexico
Omewa brace powers Falcons past Senegal
Joshua open to Itauma fight
NWFL Super Six: Edo Queens’ beat FC Robo 4-1
Rangers International going, going . . . (63,844 views)
Amaju Pinnick: A cat with nine lives (55,177 views)
Second Term: Amaju Pinnick, Other NFF Heavyweights Home to Roost •How Pinnick Broke the Jinx (53,125 views)
Current issues in Nigerian sports: Matters arising (52,693 views)
Sports Development: Zenith Bank on the zenith (52,596 views)
Missing $150,000 IAAF Grant: Solomon Dalung’s Hide and Seek game (52,494 views)
Gov. Abdullahi Ganduje’s solid footprints, commitment to sports development in Kano State (52,329 views)
NFF Presidency: Pinnick, Maigari, Ogunjobi, Okoye in Battle for Supremacy (51,895 views)
Olopade, BET9A wave of revolution in NNL (51,076 views)
Commonwealth Games 2018: Shame of Muhammadu Buhari, Solomon Dalung (49,575 views)
Ibrahimovic’s Man U exit: Whose decision is it? And in whose interest? (48,012 views)
John Mikel Obi: Segun Odegbami’s Outrageous Call! (47,454 views)